{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,1) (0,16) (0,15) (0,17) (0,2) (0,12) (0,5) (0,13) (0,4) (0,14) (0,10) (1,13) (1,17) (1,10) (1,3) (1,4) (1,2) (1,5) (1,9) (1,15) (1,7) (2,6) (2,15) (2,4) (2,8) (2,17) (2,9) (2,7) (2,14) (2,3) (2,13) (2,12) (2,5) (3,17) (3,9) (3,12) (3,6) (3,4) (3,14) (3,11) (3,8) (4,13) (4,17) (4,11) (4,10) (4,16) (4,7) (4,6) (4,8) (4,14) (5,14) (5,13) (5,12) (5,6) (5,7) (5,9) (5,17) (5,11) (6,7) (6,12) (6,10) (6,11) (6,17) (6,16) (7,17) (7,8) (7,11) (7,10) (7,9) (7,15) (7,12) (7,13) (8,16) (8,11) (8,13) (8,12) (8,9) (8,17) (9,14) (9,11) (9,17) (9,10) (9,16) (9,15) (9,13) (10,15) (10,16) (10,11) (10,14) (10,13) (11,12) (11,13) (11,15) (11,17) (11,14) (11,16) (12,17) (12,16) (12,13) (12,14) (13,16) (13,15) (14,15) (15,16) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,13,15,16,17,12,14,11,9,8,7,6,5,2,3,4,1", "difficulty": "hard", "doc_id": "0"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,3) (0,10) (0,4) (0,11) (1,12) (1,11) (1,2) (1,13) (1,9) (1,4) (2,10) (2,7) (2,8) (2,15) (2,17) (2,6) (3,10) (3,12) (3,13) (3,16) (3,15) (3,6) (3,4) (4,7) (4,12) (4,11) (4,13) (4,5) (4,15) (5,9) (5,17) (5,12) (6,15) (6,17) (6,9) (6,7) (7,11) (7,8) (7,9) (8,11) (8,14) (9,13) (9,17) (9,14) (9,11) (9,16) (10,12) (10,17) (10,14) (10,16) (11,16) (11,15) (12,13) (12,16) (13,16) (13,15) (13,14) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,15,13,14,16,12,10,17,9,5,4,3,6,7,8,2,1", "difficulty": "hard", "doc_id": "1"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,8) (0,3) (0,4) (0,11) (0,12) (1,13) (1,2) (1,12) (1,8) (2,6) (2,5) (2,9) (2,8) (2,3) (3,4) (3,12) (3,5) (3,13) (3,8) (3,11) (4,8) (4,5) (4,12) (4,6) (4,13) (5,6) (5,9) (5,13) (5,10) (6,9) (6,8) (7,8) (7,11) (7,12) (8,12) (9,10) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,11,7,8,6,9,10,5,13,4,3,2,1", "difficulty": "hard", "doc_id": "2"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,7) (0,9) (0,5) (0,10) (0,6) (0,1) (0,11) (1,3) (1,8) (1,4) (1,2) (1,6) (1,5) (2,3) (2,9) (2,8) (2,11) (2,4) (2,10) (2,6) (2,5) (3,6) (3,10) (3,7) (3,11) (3,9) (4,8) (4,7) (5,6) (5,8) (5,11) (5,9) (5,10) (6,8) (6,10) (6,9) (6,7) (7,10) (7,9) (7,11) (8,11) (8,9) (9,10) (9,11) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,10,9,8,6,7,4,2,5,1,3", "difficulty": "hard", "doc_id": "3"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,1) (0,2) (1,2) (1,5) (1,3) (2,5) (2,3) (2,6) (3,6) (3,4)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,5,2,6,3,4", "difficulty": "easy", "doc_id": "4"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (0,4) (0,7) (0,6) (0,1) (1,5) (1,8) (2,4) (2,9) (2,7) (2,6) (2,8) (3,9) (3,7) (4,6) (4,7) (5,7) (5,6) (6,8) (6,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,8,6,5,7,4,2,9,3", "difficulty": "easy", "doc_id": "5"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,15) (0,11) (0,9) (0,4) (0,10) (0,13) (0,2) (0,6) (0,5) (0,1) (0,8) (1,15) (1,2) (1,13) (1,9) (1,3) (1,12) (1,6) (1,7) (1,5) (1,8) (2,8) (2,9) (2,11) (2,15) (2,12) (2,7) (2,6) (2,5) (2,10) (2,13) (3,10) (3,7) (3,6) (3,5) (3,12) (3,11) (3,9) (4,15) (4,6) (4,8) (4,5) (4,7) (5,12) (5,11) (5,14) (5,7) (5,10) (5,8) (5,6) (6,7) (6,11) (6,14) (6,10) (6,13) (6,15) (7,11) (7,13) (7,9) (7,15) (7,8) (7,10) (8,12) (8,14) (8,9) (8,13) (8,10) (8,11) (8,15) (9,11) (9,12) (10,13) (10,12) (11,15) (11,14) (11,13) (12,15) (13,14) (13,15) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,15,14,13,11,9,12,10,7,6,4,5,3,1,2", "difficulty": "hard", "doc_id": "6"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,5) (0,2) (0,16) (0,11) (0,9) (0,12) (0,6) (0,4) (1,8) (1,5) (1,6) (1,2) (1,3) (1,14) (1,10) (1,9) (1,16) (1,4) (1,11) (1,13) (1,7) (2,11) (2,6) (2,12) (2,16) (2,14) (2,7) (2,13) (2,5) (2,8) (3,6) (3,4) (3,7) (3,10) (3,15) (3,12) (3,5) (4,15) (4,14) (4,11) (4,8) (4,5) (4,6) (4,7) (5,7) (5,6) (5,15) (5,13) (5,12) (5,11) (5,10) (6,7) (6,14) (6,12) (6,16) (6,10) (6,13) (6,11) (7,15) (7,8) (7,13) (7,11) (7,12) (7,10) (8,14) (8,15) (8,13) (8,9) (8,16) (8,11) (9,12) (9,13) (9,11) (9,14) (10,16) (10,13) (10,12) (11,13) (11,15) (11,12) (12,14) (12,13) (12,16) (13,16) (13,15) (15,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,7,10,12,16,15,13,11,9,14,8,2,5,6,3,1", "difficulty": "hard", "doc_id": "7"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,7) (0,2) (0,4) (0,14) (0,16) (0,10) (0,1) (0,3) (0,15) (1,9) (1,4) (1,11) (1,13) (1,10) (1,6) (1,14) (1,7) (1,2) (2,6) (2,10) (2,9) (2,15) (3,6) (3,17) (3,15) (3,9) (3,16) (3,5) (4,7) (4,12) (4,13) (4,18) (5,16) (5,8) (5,15) (5,9) (5,12) (5,18) (5,11) (5,13) (6,10) (6,12) (6,9) (6,14) (6,7) (6,11) (7,9) (7,10) (7,14) (7,12) (8,9) (8,15) (9,14) (10,17) (10,16) (10,12) (11,18) (12,14) (13,15) (13,18) (14,18) (14,17) (15,16) (15,18) (16,18) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,15,18,17,14,12,10,16,3,5,13,4,7,6,11,1,2,9,8", "difficulty": "hard", "doc_id": "8"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,1) (0,9) (0,6) (0,5) (0,4) (0,7) (1,12) (1,4) (1,6) (1,10) (1,9) (1,7) (1,2) (2,11) (2,3) (2,7) (2,4) (2,12) (3,5) (3,7) (3,8) (3,6) (3,11) (3,4) (3,9) (4,10) (4,11) (4,8) (4,9) (4,5) (4,12) (4,6) (5,10) (5,6) (5,9) (5,7) (6,10) (6,11) (6,12) (6,8) (7,8) (7,12) (7,9) (8,10) (8,9) (9,10) (9,11) (10,12) (10,11) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,9,11,12,10,8,6,5,4,3,2,1", "difficulty": "hard", "doc_id": "9"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,13) (0,2) (0,4) (0,11) (0,3) (0,9) (0,10) (1,7) (1,3) (1,8) (1,2) (1,6) (1,12) (1,11) (1,5) (2,9) (2,6) (2,7) (2,3) (2,4) (2,8) (3,7) (3,13) (3,5) (3,9) (3,4) (3,8) (4,6) (5,13) (5,12) (5,9) (5,6) (5,11) (6,9) (6,12) (7,12) (7,10) (7,13) (7,11) (8,10) (8,9) (8,13) (8,11) (9,11) (9,10) (10,11) (10,13) (11,13) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,13,11,12,7,3,8,9,6,5,1,2,4", "difficulty": "hard", "doc_id": "10"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,4) (0,6) (0,9) (0,10) (0,2) (0,1) (1,3) (1,4) (1,5) (1,6) (2,4) (2,9) (2,6) (3,8) (3,7) (3,4) (4,8) (5,7) (5,9) (5,8) (5,6) (6,10) (6,7) (7,8) (8,9) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,6,7,5,8,3,4,2,9,10", "difficulty": "hard", "doc_id": "11"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,11) (0,5) (0,4) (1,12) (1,8) (1,9) (1,2) (1,3) (1,11) (1,6) (1,4) (2,12) (2,6) (2,3) (2,4) (2,5) (3,9) (3,5) (3,11) (3,4) (4,12) (4,5) (4,11) (4,9) (4,6) (5,7) (5,9) (5,12) (5,11) (5,6) (6,12) (6,11) (6,7) (7,12) (7,11) (7,8) (8,12) (8,11) (9,11) (9,12) (10,12) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,6,7,8,11,9,5,3,2,1,12,10", "difficulty": "hard", "doc_id": "12"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,8) (0,1) (0,6) (0,3) (0,5) (1,6) (1,3) (1,8) (1,5) (1,2) (2,4) (2,5) (2,3) (2,6) (3,7) (3,6) (4,6) (4,5) (5,6) (5,7) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,7,6,4,2,3,1,8", "difficulty": "easy", "doc_id": "13"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,7) (0,3) (0,1) (1,3) (2,6) (2,3) (2,5) (4,7) (5,6) (5,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,3,2,6,5,7,4", "difficulty": "easy", "doc_id": "14"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,13) (0,14) (0,5) (0,9) (0,1) (0,2) (0,7) (1,5) (1,8) (1,10) (1,9) (1,11) (2,4) (2,9) (2,3) (2,7) (3,4) (3,6) (3,10) (3,7) (3,8) (3,12) (4,13) (4,14) (4,10) (5,14) (5,9) (5,8) (5,13) (6,9) (6,7) (7,11) (7,12) (7,9) (7,10) (7,8) (8,12) (8,10) (9,13) (10,12) (10,13) (11,13) (12,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,8,10,13,11,1,5,9,6,3,12,14,4,2", "difficulty": "hard", "doc_id": "15"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,8) (0,6) (0,12) (0,9) (0,11) (0,3) (0,15) (0,13) (0,7) (1,6) (1,2) (1,13) (1,15) (1,8) (1,14) (1,10) (1,17) (1,16) (1,12) (2,10) (2,6) (2,13) (2,14) (2,11) (2,12) (2,15) (3,13) (3,7) (3,10) (3,6) (3,4) (3,11) (3,14) (4,9) (4,11) (4,5) (4,13) (4,8) (5,6) (5,14) (5,11) (5,7) (5,16) (5,10) (5,13) (5,12) (5,17) (6,17) (6,8) (6,16) (6,14) (6,13) (6,10) (7,11) (7,12) (7,17) (7,16) (7,10) (7,14) (7,8) (8,9) (8,17) (8,12) (8,14) (8,11) (8,16) (8,15) (9,15) (9,11) (9,13) (9,16) (9,14) (9,10) (9,12) (10,16) (10,14) (10,17) (10,13) (11,16) (11,14) (11,15) (12,13) (12,15) (13,17) (13,15) (13,16) (14,15) (14,17) (15,17) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,8,15,17,16,13,12,9,10,14,11,5,4,3,6,2,1", "difficulty": "hard", "doc_id": "16"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,4) (0,5) (0,1) (1,6) (1,5) (1,4) (2,5) (2,3) (2,4) (3,5) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,4,2,3,5,6", "difficulty": "easy", "doc_id": "17"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,5) (0,8) (0,9) (1,8) (1,3) (1,6) (2,4) (2,6) (2,8) (3,4) (3,5) (3,8) (3,7) (4,8) (4,7) (4,9) (5,6) (5,8) (6,9) (6,8) (6,7) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,7,6,5,8,2,4,3,1", "difficulty": "easy", "doc_id": "18"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,10) (0,5) (0,2) (0,8) (0,1) (0,16) (0,13) (1,12) (1,5) (1,9) (1,6) (2,11) (2,15) (2,16) (2,8) (2,14) (3,9) (3,15) (3,6) (4,15) (4,11) (4,16) (4,10) (4,14) (4,7) (4,13) (5,7) (5,9) (5,14) (5,13) (6,11) (6,16) (6,15) (6,8) (7,9) (7,14) (7,16) (8,13) (8,10) (8,9) (8,16) (9,11) (9,13) (10,12) (11,15) (11,16) (12,16) (12,14) (12,13) (14,16) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,13,12,14,15,11,16,6,3,9,1,5,7,4,10,8,2", "difficulty": "hard", "doc_id": "19"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,9) (1,5) (1,10) (1,3) (1,2) (2,5) (2,7) (2,8) (2,11) (2,6) (2,12) (2,10) (3,8) (3,11) (4,5) (4,8) (4,10) (4,9) (4,12) (5,8) (5,7) (5,11) (6,12) (6,8) (7,12) (7,8) (8,11) (9,10) (9,11) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,11,10,4,12,7,8,6,2,5,1,3", "difficulty": "hard", "doc_id": "20"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (0,8) (0,3) (0,2) (1,8) (1,7) (1,2) (1,3) (1,4) (2,7) (2,8) (2,3) (2,6) (3,5) (3,6) (3,9) (3,7) (3,8) (3,4) (4,6) (4,9) (4,8) (5,9) (5,8) (5,7) (5,6) (6,8) (6,7) (7,9) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,6,7,8,9,5,3,4,1", "difficulty": "easy", "doc_id": "21"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,1) (0,4) (0,13) (1,15) (1,5) (1,7) (1,4) (1,6) (2,3) (2,12) (2,13) (2,7) (2,9) (2,15) (3,15) (3,10) (3,11) (3,9) (3,4) (4,11) (4,13) (4,12) (4,7) (5,15) (5,13) (5,11) (5,9) (5,6) (6,9) (6,13) (6,10) (6,11) (6,7) (7,8) (7,14) (7,15) (7,9) (7,13) (7,11) (7,12) (8,11) (8,14) (9,12) (9,11) (9,14) (9,10) (10,11) (10,12) (10,13) (11,13) (12,15) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,13,14,9,10,12,15,7,8,11,6,5,1,4,3,2", "difficulty": "hard", "doc_id": "22"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,8) (0,6) (0,4) (0,10) (1,7) (1,4) (1,6) (1,8) (1,3) (1,9) (1,10) (2,10) (2,9) (2,6) (2,8) (3,6) (4,7) (4,6) (4,5) (5,6) (5,8) (6,8) (6,9) (6,7) (8,9) (8,10) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,9,8,5,4,7,1,3,6,2", "difficulty": "hard", "doc_id": "23"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,7) (0,8) (0,4) (1,4) (1,7) (1,5) (1,3) (1,2) (1,8) (2,7) (2,8) (2,4) (3,6) (3,8) (3,7) (4,5) (4,7) (4,6) (4,8) (5,7) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,8,6,3,7,5,1,2", "difficulty": "easy", "doc_id": "24"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,13) (0,14) (0,1) (0,2) (0,12) (0,16) (0,4) (0,3) (0,9) (0,6) (1,13) (1,2) (1,5) (1,3) (1,14) (1,4) (1,15) (1,16) (1,11) (1,12) (1,8) (1,7) (2,8) (2,4) (2,14) (2,10) (2,5) (2,12) (2,9) (2,15) (3,12) (3,8) (3,11) (3,13) (3,15) (3,10) (3,14) (3,4) (4,15) (4,7) (4,10) (4,8) (4,12) (4,6) (5,7) (5,13) (5,12) (5,11) (5,16) (5,10) (5,6) (5,8) (6,10) (6,9) (6,14) (6,8) (6,13) (6,7) (6,11) (7,15) (7,8) (7,9) (7,13) (7,10) (8,16) (8,11) (8,15) (8,12) (8,10) (8,9) (9,13) (9,12) (9,11) (9,15) (10,12) (10,14) (11,14) (12,13) (12,16) (13,16) (13,14) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,11,14,16,13,12,10,8,9,15,7,5,2,4,3,1", "difficulty": "hard", "doc_id": "25"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,7) (0,14) (0,8) (0,9) (0,1) (0,12) (0,5) (0,11) (0,4) (1,4) (1,6) (1,12) (2,3) (2,7) (2,8) (2,5) (2,4) (2,9) (2,13) (2,14) (2,6) (3,5) (3,10) (3,4) (3,7) (3,13) (3,8) (3,9) (3,14) (4,9) (4,14) (4,7) (4,5) (4,11) (4,6) (5,10) (5,13) (5,12) (5,7) (5,11) (6,10) (6,8) (6,11) (6,7) (6,9) (6,14) (7,13) (7,9) (7,14) (7,10) (7,12) (8,13) (8,10) (8,9) (8,12) (8,14) (9,10) (9,11) (9,12) (9,13) (9,14) (10,11) (11,13) (11,14) (11,12) (12,14) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,6,14,11,13,9,10,8,3,7,2,5,12,1", "difficulty": "hard", "doc_id": "26"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,10) (0,5) (0,4) (0,9) (0,2) (0,8) (0,1) (0,6) (1,3) (1,2) (1,10) (2,7) (2,3) (2,5) (2,10) (2,8) (2,6) (2,9) (3,10) (3,5) (4,6) (4,8) (4,5) (5,6) (5,10) (5,7) (5,9) (5,8) (6,7) (6,10) (7,9) (7,8) (7,10) (8,10) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,10,9,7,8,4,5,3,2,1", "difficulty": "hard", "doc_id": "27"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,3) (0,2) (0,4) (1,4) (1,2) (2,3) (2,5) (2,6) (3,5) (4,6) (4,7) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,5,6,2,1,4,7", "difficulty": "easy", "doc_id": "28"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,6) (0,5) (0,3) (1,5) (1,3) (1,4) (1,2) (2,6) (2,3) (3,4) (3,6) (3,5) (4,6) (4,5) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,5,6,4,1,2", "difficulty": "easy", "doc_id": "29"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,5) (0,9) (0,8) (0,3) (0,2) (0,6) (1,3) (1,7) (1,6) (1,4) (1,2) (1,5) (2,9) (2,6) (2,4) (3,7) (3,9) (4,8) (4,6) (5,9) (5,8) (5,6) (6,9) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,9,5,8,7,3,1,2,4", "difficulty": "easy", "doc_id": "30"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,1) (0,4) (0,14) (0,19) (0,8) (0,17) (0,3) (0,13) (1,9) (1,8) (1,18) (1,6) (1,13) (1,19) (1,15) (1,12) (1,16) (1,2) (1,10) (1,3) (2,18) (2,15) (2,14) (2,8) (2,13) (2,5) (2,19) (3,19) (3,6) (3,11) (3,10) (3,18) (4,17) (4,14) (4,12) (4,5) (4,16) (4,9) (5,8) (5,12) (5,7) (5,19) (5,13) (6,17) (6,9) (6,19) (6,11) (6,12) (7,9) (7,18) (7,14) (7,13) (7,17) (7,19) (8,18) (8,19) (8,15) (8,13) (9,16) (9,18) (9,10) (9,14) (10,17) (11,17) (11,12) (11,15) (12,13) (12,19) (13,16) (13,18) (13,19) (13,14) (14,19) (15,19) (15,16) (15,18) (16,19) (16,17) (17,18) (18,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,13,14,19,18,17,16,15,11,12,6,3,10,1,2,8,5,7,9,4", "difficulty": "hard", "doc_id": "31"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,6) (0,10) (0,1) (1,9) (1,5) (1,2) (1,7) (2,3) (3,8) (3,11) (3,4) (4,10) (4,9) (4,6) (5,8) (5,7) (5,9) (6,10) (6,7) (7,10) (7,9) (8,9) (8,11) (9,11) (9,10) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,7,9,5,8,11,10,6,4,3,2", "difficulty": "hard", "doc_id": "32"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,5) (0,6) (0,1) (1,7) (1,5) (1,3) (1,6) (2,7) (2,5) (2,3) (3,6) (4,5) (4,7) (5,7) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,6,5,4,7,2,3", "difficulty": "easy", "doc_id": "33"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,1) (0,2) (1,12) (1,3) (1,4) (1,6) (2,11) (2,12) (2,4) (2,5) (2,8) (2,3) (2,9) (2,6) (3,10) (3,6) (3,5) (3,7) (4,7) (4,12) (4,8) (5,11) (5,12) (5,10) (6,8) (6,11) (7,10) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,6,11,5,10,3,7,8,4,12,2,9", "difficulty": "hard", "doc_id": "34"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,5) (0,3) (0,2) (0,4) (0,6) (1,2) (1,3) (1,4) (2,6) (2,4) (2,5) (3,5) (3,7) (3,6) (4,7) (5,7) (5,6) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,7,5,3,1,4,2", "difficulty": "easy", "doc_id": "35"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,4) (0,7) (0,15) (0,9) (0,3) (0,12) (0,8) (0,1) (0,16) (0,10) (0,2) (1,13) (1,8) (1,7) (1,12) (1,4) (1,10) (1,9) (1,16) (1,5) (1,3) (2,8) (2,9) (2,3) (2,6) (2,4) (2,17) (2,16) (2,7) (3,6) (3,10) (3,13) (3,17) (3,8) (3,15) (3,11) (3,7) (3,9) (3,4) (4,13) (4,14) (4,12) (4,10) (4,6) (5,16) (5,10) (5,15) (5,17) (5,9) (5,12) (5,13) (6,11) (6,7) (6,8) (6,13) (6,14) (6,12) (6,16) (6,15) (7,15) (7,10) (7,13) (7,16) (7,12) (8,11) (8,9) (8,17) (8,14) (8,16) (8,13) (9,14) (9,10) (9,17) (9,15) (10,13) (10,14) (10,12) (10,15) (11,13) (11,16) (11,15) (12,16) (12,17) (12,13) (12,14) (13,14) (13,15) (14,16) (15,16) (15,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,7,12,14,16,15,17,9,10,13,11,8,6,4,3,1,5", "difficulty": "hard", "doc_id": "36"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,2) (0,4) (0,7) (0,9) (1,9) (1,4) (1,10) (2,10) (2,5) (2,4) (3,10) (3,8) (4,9) (4,7) (5,6) (5,7) (5,9) (5,8) (5,10) (6,10) (6,7) (8,10) (8,9) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,8,3,10,6,7,5,2,4,1", "difficulty": "hard", "doc_id": "37"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,4) (0,6) (0,8) (0,7) (0,3) (0,9) (0,2) (0,11) (1,9) (1,5) (1,8) (1,6) (2,3) (2,5) (2,11) (2,8) (3,4) (3,10) (3,9) (3,11) (4,6) (4,7) (4,5) (4,8) (5,8) (5,11) (5,7) (6,10) (6,8) (6,11) (7,9) (7,11) (7,8) (9,10) (9,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,9,10,6,8,7,4,3,2,5,1", "difficulty": "hard", "doc_id": "38"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,8) (0,16) (0,15) (0,19) (0,18) (0,5) (0,7) (0,14) (0,1) (0,17) (0,2) (1,9) (1,13) (1,18) (1,6) (1,17) (1,12) (1,15) (1,2) (1,10) (1,4) (1,16) (2,8) (2,10) (2,18) (2,17) (2,15) (2,6) (2,19) (2,12) (2,7) (2,16) (2,9) (2,5) (3,6) (3,11) (3,4) (3,13) (3,16) (3,19) (3,15) (3,17) (4,19) (4,17) (4,14) (4,18) (4,10) (4,13) (4,8) (4,6) (4,7) (4,5) (5,6) (5,8) (5,19) (5,15) (5,13) (5,16) (5,14) (5,7) (6,18) (6,9) (6,14) (6,19) (6,11) (7,15) (7,13) (7,9) (7,11) (7,12) (7,8) (8,9) (8,12) (8,19) (8,13) (8,14) (8,18) (8,11) (9,14) (9,13) (9,11) (9,19) (9,16) (9,18) (10,14) (10,11) (10,12) (10,19) (10,16) (10,17) (10,15) (10,13) (10,18) (11,16) (11,14) (11,19) (11,18) (11,12) (12,17) (12,18) (12,15) (12,13) (12,16) (12,19) (12,14) (13,14) (13,15) (13,17) (13,18) (14,19) (14,17) (14,15) (15,16) (16,19) (16,17) (17,19) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,5,7,8,11,12,14,15,16,17,18,13,10,19,9,6,3,4,1", "difficulty": "hard", "doc_id": "39"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,3) (0,12) (0,7) (0,4) (0,5) (0,6) (1,11) (1,10) (1,2) (2,4) (2,6) (2,5) (2,12) (2,7) (2,8) (2,11) (3,11) (3,9) (3,6) (4,12) (4,6) (4,11) (4,9) (5,6) (5,10) (5,9) (6,12) (6,7) (7,12) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,7,12,4,9,8,2,5,10,1,11,3", "difficulty": "hard", "doc_id": "40"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,5) (0,2) (0,4) (1,4) (1,6) (1,3) (1,7) (1,2) (2,3) (3,6) (3,7) (4,5) (4,7) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,3,7,6,1,4,5", "difficulty": "easy", "doc_id": "41"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,14) (0,10) (0,11) (0,8) (0,15) (0,5) (0,4) (1,7) (1,5) (1,8) (1,13) (2,5) (2,3) (3,13) (3,5) (3,11) (4,15) (4,13) (4,14) (4,7) (4,9) (5,6) (5,7) (5,9) (5,12) (5,10) (6,8) (6,13) (6,7) (6,11) (6,10) (6,9) (6,15) (7,10) (8,12) (8,14) (8,13) (8,10) (8,9) (9,13) (10,11) (10,12) (10,14) (11,13) (11,15) (11,12) (12,15) (13,15) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,9,13,14,10,12,15,11,6,8,1,7,5,3,2", "difficulty": "hard", "doc_id": "42"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,3) (0,7) (0,16) (0,1) (0,13) (0,18) (0,17) (1,14) (1,18) (1,10) (1,7) (1,13) (1,17) (1,5) (1,3) (1,4) (2,13) (2,12) (2,9) (2,4) (2,15) (2,8) (2,5) (2,3) (3,17) (3,19) (3,4) (3,9) (3,7) (3,11) (3,13) (3,5) (3,12) (4,19) (4,9) (4,10) (4,17) (4,8) (5,13) (5,6) (5,12) (5,19) (5,10) (6,13) (6,8) (6,16) (6,17) (6,10) (7,16) (7,17) (7,18) (7,13) (7,8) (8,9) (8,11) (8,13) (8,19) (8,16) (9,19) (9,17) (9,15) (9,11) (9,10) (10,16) (10,15) (10,18) (11,19) (11,12) (11,15) (12,18) (12,15) (12,19) (13,17) (13,18) (13,16) (14,16) (14,17) (14,19) (15,19) (16,19) (17,19) (18,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,17,19,18,13,16,14,1,4,8,11,15,12,5,6,10,9,2,3,7", "difficulty": "hard", "doc_id": "43"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,7) (0,9) (0,2) (0,8) (0,4) (0,3) (1,7) (1,8) (1,6) (1,9) (1,4) (2,9) (2,8) (2,6) (3,5) (4,6) (4,7) (5,7) (5,6) (6,8) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,5,6,8,7,4,1,9,2", "difficulty": "easy", "doc_id": "44"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,10) (0,5) (0,1) (0,9) (0,6) (0,4) (0,3) (1,4) (1,10) (1,2) (1,6) (1,7) (2,9) (2,6) (2,3) (3,8) (3,9) (4,8) (4,9) (4,7) (4,5) (5,10) (5,9) (6,8) (6,10) (7,8) (7,10) (7,9) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,9,10,7,8,6,2,1,4,5", "difficulty": "hard", "doc_id": "45"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,2) (0,5) (0,6) (0,9) (0,8) (0,3) (1,4) (1,6) (1,5) (1,9) (2,8) (2,3) (3,9) (3,6) (3,8) (3,4) (5,8) (6,9) (6,8) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,8,2,3,4,1,6,9,7", "difficulty": "easy", "doc_id": "46"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,3) (0,11) (0,5) (0,4) (0,6) (0,8) (0,12) (1,14) (1,3) (1,9) (1,8) (2,3) (2,9) (2,6) (2,7) (2,5) (2,4) (3,8) (3,11) (3,7) (3,5) (3,14) (3,12) (4,10) (4,7) (4,8) (4,13) (4,9) (4,14) (4,5) (5,12) (5,8) (5,14) (5,9) (5,13) (6,8) (6,9) (6,13) (6,10) (6,11) (6,7) (7,14) (7,12) (7,9) (7,8) (7,10) (7,13) (8,14) (8,13) (8,9) (9,12) (9,10) (10,14) (11,13) (12,14) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,14,13,11,6,7,10,9,8,5,4,2,3,1", "difficulty": "hard", "doc_id": "47"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,1) (0,6) (0,5) (1,2) (1,4) (2,5) (3,8) (3,4) (3,7) (4,7) (4,6) (5,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,4,3,8,7,5,2,1", "difficulty": "easy", "doc_id": "48"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,1) (0,7) (0,6) (1,7) (1,6) (2,3) (2,7) (2,6) (2,5) (3,6) (3,5) (4,6) (4,5) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,4,5,3,2,7,1", "difficulty": "easy", "doc_id": "49"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,7) (0,5) (0,2) (0,3) (0,4) (1,4) (2,3) (2,4) (2,6) (2,8) (3,8) (3,6) (3,5) (4,7) (5,8) (5,6) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,5,6,2,8,7,4,1", "difficulty": "easy", "doc_id": "50"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,7) (0,5) (0,10) (0,9) (0,3) (0,2) (0,1) (1,2) (1,4) (1,3) (1,10) (1,6) (1,7) (2,4) (2,10) (2,6) (2,7) (2,5) (3,4) (3,9) (3,5) (4,9) (4,8) (4,6) (4,7) (4,5) (6,7) (6,10) (6,9) (7,10) (7,9) (8,9) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,7,9,10,6,2,5,3,4,8", "difficulty": "hard", "doc_id": "51"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,6) (0,4) (1,5) (1,2) (1,3) (2,4) (2,6) (3,4) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,2,4,5,1,3", "difficulty": "easy", "doc_id": "52"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (0,1) (0,9) (0,4) (0,5) (0,6) (0,7) (1,5) (1,4) (1,9) (1,7) (1,8) (2,6) (2,9) (2,4) (2,7) (2,3) (2,5) (3,8) (3,5) (3,9) (3,4) (3,7) (4,8) (4,7) (5,8) (5,9) (6,8) (6,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,4,8,6,9,3,2,5,1", "difficulty": "easy", "doc_id": "53"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,2) (0,7) (0,1) (0,5) (0,6) (1,2) (1,6) (1,5) (2,4) (2,7) (2,6) (3,6) (3,5) (3,4) (3,7) (4,5) (5,7) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,5,7,3,4,2,1", "difficulty": "easy", "doc_id": "54"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (0,9) (0,1) (0,6) (1,6) (1,8) (2,8) (2,7) (2,4) (2,6) (2,3) (3,8) (3,6) (3,9) (3,7) (3,5) (4,8) (4,6) (4,7) (5,6) (5,8) (5,9) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,5,9,7,4,2,3,8,1", "difficulty": "easy", "doc_id": "55"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,7) (0,11) (0,4) (0,6) (0,9) (0,12) (0,5) (1,4) (1,10) (1,6) (1,7) (1,8) (1,9) (1,12) (2,4) (2,9) (2,7) (2,6) (2,8) (2,10) (2,3) (2,5) (3,11) (3,6) (3,9) (3,5) (3,8) (3,7) (3,12) (4,7) (4,10) (4,12) (4,6) (5,6) (5,8) (5,7) (6,8) (6,11) (7,9) (7,8) (7,11) (8,10) (8,9) (8,12) (9,12) (9,10) (9,11) (10,12) (10,11) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,7,11,12,10,9,8,6,3,2,4,1", "difficulty": "hard", "doc_id": "56"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,1) (0,8) (0,6) (1,3) (1,9) (2,3) (2,9) (2,4) (3,4) (3,5) (3,7) (3,8) (4,8) (5,7) (5,9) (6,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,8,4,2,9,5,7,3,1", "difficulty": "easy", "doc_id": "57"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,7) (0,17) (0,16) (0,1) (0,9) (0,2) (0,15) (0,4) (0,13) (0,8) (0,11) (0,14) (0,10) (1,10) (1,4) (1,17) (1,15) (1,2) (1,14) (1,9) (1,6) (1,12) (1,16) (2,16) (2,8) (2,9) (2,12) (2,13) (2,5) (2,7) (2,6) (3,12) (3,6) (3,16) (3,8) (3,13) (3,7) (4,16) (4,17) (4,7) (4,12) (4,13) (5,8) (5,9) (5,16) (5,13) (5,17) (5,7) (5,15) (6,11) (6,14) (6,12) (6,8) (6,13) (6,17) (6,16) (6,9) (7,10) (7,14) (7,13) (7,17) (7,15) (7,8) (7,9) (8,15) (8,13) (8,9) (8,16) (8,12) (8,17) (9,17) (9,10) (9,15) (9,13) (9,11) (10,13) (10,11) (10,15) (10,14) (11,14) (11,12) (11,13) (11,16) (12,13) (12,16) (13,16) (13,15) (13,14) (13,17) (14,16) (15,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,14,16,15,13,17,9,11,12,8,6,3,7,5,2,1,4", "difficulty": "hard", "doc_id": "58"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,9) (0,2) (0,6) (0,1) (0,3) (0,12) (0,14) (0,8) (1,4) (1,13) (1,5) (1,2) (1,14) (1,9) (1,6) (1,11) (2,11) (2,3) (2,6) (2,8) (2,12) (2,13) (2,14) (2,7) (3,14) (3,6) (3,11) (4,14) (4,5) (4,11) (4,6) (5,12) (5,8) (5,10) (6,11) (6,14) (6,12) (6,13) (7,10) (7,11) (7,13) (8,9) (8,11) (8,13) (8,12) (9,10) (9,12) (9,13) (10,13) (11,13) (12,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,12,14,6,13,11,4,5,1,9,10,7,2,3", "difficulty": "hard", "doc_id": "59"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,10) (0,2) (0,4) (0,6) (1,4) (1,10) (1,9) (1,8) (1,5) (1,7) (2,4) (2,10) (2,8) (2,5) (3,9) (3,8) (3,6) (3,10) (4,7) (4,9) (4,5) (5,7) (6,7) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,7,9,8,3,10,2,5,4,1", "difficulty": "hard", "doc_id": "60"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,16) (0,8) (0,1) (0,17) (0,10) (0,3) (0,12) (0,7) (0,5) (1,16) (1,9) (1,4) (1,3) (1,2) (1,13) (1,6) (2,15) (2,8) (2,14) (2,5) (2,11) (3,9) (3,8) (3,13) (3,6) (3,15) (3,11) (3,10) (4,13) (4,14) (4,16) (4,15) (5,10) (5,7) (5,11) (5,16) (5,12) (5,8) (5,14) (5,17) (6,14) (6,9) (6,10) (6,8) (7,17) (7,14) (7,12) (7,9) (7,16) (8,16) (8,12) (8,10) (8,11) (8,9) (9,16) (9,13) (10,14) (10,11) (10,15) (10,13) (11,13) (11,14) (11,12) (11,17) (12,14) (12,16) (13,15) (14,16) (14,17) (15,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,17,14,16,15,13,11,12,7,9,6,10,3,8,2,1,4", "difficulty": "hard", "doc_id": "61"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,5) (0,7) (0,1) (0,4) (1,2) (1,9) (1,8) (2,6) (2,5) (2,4) (2,9) (3,8) (3,6) (4,8) (4,9) (5,9) (5,8) (5,7) (5,6) (6,7) (6,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,9,8,3,6,7,5,2,1", "difficulty": "easy", "doc_id": "62"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (0,2) (0,8) (0,7) (1,4) (1,9) (1,3) (1,2) (1,6) (2,7) (2,6) (2,8) (3,5) (4,8) (5,7) (7,9) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,7,5,3,1,6,2,8,4", "difficulty": "easy", "doc_id": "63"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,12) (0,15) (0,10) (0,1) (0,5) (0,8) (0,2) (0,11) (0,13) (0,3) (0,7) (0,19) (0,9) (1,4) (1,6) (1,3) (1,9) (1,11) (1,18) (1,2) (1,13) (1,19) (1,16) (1,14) (1,17) (1,7) (1,15) (2,18) (2,16) (2,14) (2,3) (2,7) (2,8) (2,11) (2,5) (2,6) (2,19) (2,10) (2,15) (3,7) (3,10) (3,12) (3,9) (3,4) (3,16) (3,11) (3,14) (3,8) (4,5) (4,16) (4,10) (4,12) (4,8) (4,11) (4,6) (4,18) (4,7) (4,15) (5,9) (5,16) (5,14) (5,15) (5,13) (5,8) (5,17) (5,19) (5,7) (5,11) (6,16) (6,15) (6,14) (6,10) (6,8) (6,9) (6,17) (6,13) (6,19) (6,12) (7,13) (7,11) (7,17) (7,12) (7,15) (7,10) (8,16) (8,15) (8,14) (8,9) (8,18) (8,13) (8,11) (8,12) (9,15) (9,11) (9,13) (9,18) (9,12) (9,17) (9,19) (10,11) (10,19) (10,14) (10,16) (10,17) (10,12) (10,15) (11,18) (11,17) (11,14) (11,12) (11,15) (11,19) (11,13) (12,17) (12,16) (12,14) (12,15) (12,13) (13,18) (13,17) (13,19) (13,16) (14,15) (14,19) (14,18) (14,16) (14,17) (15,18) (16,18) (16,17) (17,18) (17,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,19,17,18,16,14,15,12,13,11,10,7,5,8,6,4,3,2,1", "difficulty": "hard", "doc_id": "64"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,8) (0,6) (0,1) (1,9) (1,7) (2,5) (2,8) (2,7) (3,8) (3,4) (3,5) (4,9) (4,8) (4,7) (5,9) (5,8) (6,7) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,9,5,3,4,8,2,7,6", "difficulty": "easy", "doc_id": "65"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (1,7) (1,6) (1,5) (2,3) (2,9) (2,6) (2,4) (3,8) (3,6) (3,5) (3,7) (4,8) (4,9) (4,7) (5,8) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,4,7,6,2,3,8,5,1", "difficulty": "easy", "doc_id": "66"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (0,4) (0,6) (0,2) (0,8) (0,7) (1,8) (1,5) (1,3) (1,7) (1,6) (1,4) (2,6) (2,5) (2,4) (2,8) (3,6) (3,4) (3,5) (4,6) (4,9) (5,9) (7,8) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,9,5,3,4,6,2,8,1", "difficulty": "easy", "doc_id": "67"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,5) (0,6) (0,9) (0,2) (1,3) (1,6) (1,9) (1,2) (2,9) (2,5) (2,6) (2,3) (3,8) (3,4) (3,7) (3,9) (4,8) (4,9) (4,5) (5,8) (5,9) (5,6) (6,8) (6,7) (7,9) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,3,9,7,8,4,5,6,1", "difficulty": "easy", "doc_id": "68"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,1) (0,3) (0,4) (0,13) (0,10) (0,11) (0,9) (1,2) (1,13) (1,8) (1,11) (1,4) (1,6) (1,10) (1,7) (1,9) (1,5) (2,11) (2,5) (2,13) (2,3) (2,4) (2,12) (2,6) (3,6) (3,11) (3,5) (3,9) (3,12) (3,10) (3,8) (4,5) (4,7) (4,10) (4,13) (5,9) (5,6) (5,7) (5,8) (6,13) (6,10) (6,7) (7,8) (7,11) (7,12) (8,13) (8,12) (8,10) (8,11) (9,12) (10,11) (10,13) (10,12) (11,12) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,12,13,10,11,8,7,6,5,4,1,2,3", "difficulty": "hard", "doc_id": "69"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,11) (0,4) (0,6) (0,7) (0,2) (0,1) (0,5) (0,3) (1,8) (1,4) (1,7) (1,10) (1,3) (1,5) (2,3) (2,11) (3,11) (3,6) (4,5) (4,9) (4,6) (5,10) (6,11) (7,10) (7,8) (8,9) (8,11) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,6,4,9,8,7,1,5,10,11,2", "difficulty": "hard", "doc_id": "70"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,11) (0,6) (0,8) (0,5) (0,16) (0,19) (0,15) (0,14) (0,17) (1,3) (1,4) (1,7) (1,2) (1,16) (1,19) (1,18) (1,9) (1,14) (1,17) (1,6) (1,13) (2,5) (2,13) (2,8) (2,11) (2,10) (2,15) (2,6) (2,7) (2,17) (2,3) (2,12) (2,18) (3,14) (3,8) (3,11) (3,17) (3,4) (3,5) (3,9) (3,13) (3,10) (3,18) (4,15) (4,11) (4,8) (4,19) (4,12) (4,14) (4,5) (4,13) (5,17) (5,13) (5,12) (5,18) (5,6) (5,9) (5,15) (5,8) (5,7) (5,19) (5,14) (5,10) (5,16) (6,8) (6,18) (6,16) (6,14) (6,13) (6,15) (7,10) (7,16) (7,9) (7,11) (7,8) (7,18) (7,17) (8,19) (8,15) (8,17) (8,14) (8,12) (8,18) (8,10) (9,16) (9,19) (9,11) (9,15) (9,10) (10,18) (10,12) (10,17) (10,13) (10,14) (10,16) (11,17) (11,14) (11,15) (12,18) (12,19) (13,19) (13,17) (13,15) (13,18) (13,14) (14,18) (14,16) (14,17) (14,15) (15,16) (15,18) (16,18) (16,17) (17,19) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,17,18,16,15,14,13,19,12,10,9,11,7,8,6,5,4,3,2,1", "difficulty": "hard", "doc_id": "71"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (0,2) (0,9) (0,7) (0,1) (1,8) (1,5) (1,2) (2,6) (3,6) (3,4) (3,9) (3,8) (4,6) (4,8) (5,9) (6,9) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,6,4,8,3,9,5,1,2", "difficulty": "easy", "doc_id": "72"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (0,5) (0,1) (0,3) (0,6) (1,3) (1,7) (1,2) (1,4) (1,9) (1,6) (1,8) (1,5) (2,7) (2,8) (2,6) (2,9) (3,5) (3,4) (3,7) (3,6) (4,9) (4,6) (5,8) (6,8) (6,9) (7,8) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,9,7,8,5,3,4,1,2", "difficulty": "easy", "doc_id": "73"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,1) (0,8) (0,5) (0,6) (0,3) (1,4) (1,8) (1,7) (1,3) (2,8) (2,5) (2,7) (2,6) (2,4) (3,6) (3,8) (4,7) (4,8) (5,6) (6,7) (6,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,8,6,7,1,4,2,5", "difficulty": "easy", "doc_id": "74"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,7) (0,6) (0,9) (0,5) (0,8) (0,17) (0,11) (0,14) (0,12) (0,15) (0,10) (1,16) (1,3) (1,10) (1,5) (1,7) (1,13) (1,12) (1,14) (1,11) (2,9) (2,12) (2,15) (2,14) (2,4) (2,11) (2,10) (2,17) (2,6) (3,16) (3,7) (3,12) (3,17) (3,11) (4,5) (4,16) (4,8) (4,7) (5,11) (5,17) (5,8) (5,16) (5,15) (5,7) (5,10) (6,12) (6,7) (6,17) (6,13) (6,15) (7,17) (7,15) (7,9) (7,16) (7,14) (8,11) (8,14) (8,16) (9,13) (9,16) (9,11) (9,12) (9,10) (10,15) (10,11) (11,12) (11,13) (11,16) (11,15) (12,14) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,11,15,14,12,9,16,8,5,7,17,3,1,13,6,2,4", "difficulty": "hard", "doc_id": "75"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,2) (0,9) (0,5) (0,10) (0,8) (0,3) (0,1) (1,6) (1,3) (1,10) (1,2) (1,9) (2,3) (2,10) (2,4) (3,5) (3,6) (3,7) (4,7) (5,10) (5,7) (5,8) (7,8) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,9,10,5,8,7,4,2,3,6", "difficulty": "hard", "doc_id": "76"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,9) (0,3) (0,1) (0,15) (0,7) (0,13) (0,5) (0,2) (0,4) (1,8) (1,13) (1,7) (1,3) (1,12) (1,15) (1,11) (1,5) (2,8) (2,10) (2,11) (2,6) (2,5) (2,4) (2,12) (2,14) (2,15) (2,13) (3,11) (3,8) (3,5) (3,7) (3,12) (3,14) (3,6) (4,10) (4,7) (4,8) (4,9) (4,5) (4,12) (5,7) (5,15) (5,6) (5,9) (5,13) (5,8) (5,12) (6,13) (6,14) (6,10) (6,7) (6,12) (6,11) (6,8) (7,8) (7,14) (7,9) (7,13) (7,12) (8,15) (8,14) (8,11) (8,13) (9,11) (9,10) (9,13) (10,11) (10,13) (10,14) (10,12) (11,15) (12,14) (13,14) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,12,14,15,11,10,13,9,7,8,6,3,1,5,2", "difficulty": "hard", "doc_id": "77"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,6) (0,5) (0,8) (0,7) (1,5) (1,6) (1,3) (2,5) (2,3) (2,6) (3,5) (3,4) (3,8) (5,6) (5,7) (5,8) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,8,5,2,6,1,3,4", "difficulty": "easy", "doc_id": "78"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,1) (0,6) (0,8) (0,3) (0,7) (1,4) (1,8) (1,7) (1,5) (2,5) (2,8) (2,7) (2,6) (2,3) (3,4) (3,7) (3,8) (3,6) (4,5) (5,6) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,8,3,6,2,5,4,1", "difficulty": "easy", "doc_id": "79"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,3) (0,1) (0,5) (1,6) (1,4) (1,7) (1,5) (1,2) (2,6) (2,4) (2,5) (3,5) (3,7) (3,4) (3,6) (4,6) (4,7) (4,5) (5,7) (5,6) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,6,7,3,4,2,1", "difficulty": "easy", "doc_id": "80"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (0,2) (0,7) (1,4) (1,8) (1,2) (1,3) (1,9) (1,7) (1,5) (2,5) (2,8) (2,4) (2,6) (2,3) (3,9) (3,5) (3,6) (3,8) (3,7) (4,5) (4,8) (4,7) (5,6) (5,8) (5,7) (6,9) (6,7) (7,8) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,9,6,5,8,4,2,3,1", "difficulty": "easy", "doc_id": "81"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,6) (0,4) (0,9) (0,2) (0,5) (1,6) (1,3) (1,11) (1,4) (2,5) (2,12) (2,11) (3,12) (3,9) (3,6) (3,11) (3,5) (4,12) (4,5) (5,10) (5,7) (5,8) (5,6) (6,11) (7,12) (7,11) (7,8) (8,12) (9,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,11,9,3,6,1,4,12,8,7,5,10", "difficulty": "hard", "doc_id": "82"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,8) (0,1) (0,5) (0,6) (1,3) (1,7) (1,8) (1,4) (2,8) (2,6) (2,3) (3,6) (4,7) (5,8) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,7,4,1,3,2,8,5", "difficulty": "easy", "doc_id": "83"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,2) (0,13) (0,7) (0,6) (0,8) (0,1) (1,9) (1,7) (1,11) (1,2) (1,8) (1,10) (1,6) (2,4) (2,7) (2,10) (2,14) (2,3) (2,13) (2,9) (3,4) (3,7) (3,9) (3,5) (4,8) (4,12) (5,9) (5,8) (5,10) (6,10) (6,8) (6,14) (6,13) (6,11) (6,12) (7,13) (7,12) (7,14) (7,11) (8,12) (9,11) (10,11) (10,12) (11,14) (11,12) (11,13) (12,13) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,6,12,13,14,11,10,5,8,4,3,9,2,7", "difficulty": "hard", "doc_id": "84"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,2) (0,3) (0,7) (0,6) (1,9) (1,10) (1,5) (2,6) (2,8) (2,7) (2,4) (2,3) (3,7) (3,10) (3,4) (4,5) (4,9) (5,7) (6,9) (6,8) (7,8) (8,9) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,8,9,10,3,4,2,7,5,1", "difficulty": "hard", "doc_id": "85"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,12) (0,2) (0,11) (0,7) (0,9) (0,4) (0,3) (0,6) (1,9) (1,6) (1,3) (1,8) (1,7) (2,4) (2,12) (2,6) (2,7) (2,10) (2,3) (3,7) (3,5) (3,12) (3,8) (3,10) (4,11) (4,8) (4,5) (4,6) (4,12) (5,11) (5,10) (5,6) (5,8) (5,7) (6,7) (6,8) (6,10) (7,9) (7,12) (7,10) (7,8) (8,9) (8,12) (9,11) (9,12) (10,12) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,10,11,9,12,8,7,5,4,2,3,1", "difficulty": "hard", "doc_id": "86"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,6) (0,3) (0,4) (0,7) (1,5) (1,2) (1,3) (2,5) (2,4) (2,6) (3,6) (3,4) (4,6) (4,7) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,4,6,5,2,1,3", "difficulty": "easy", "doc_id": "87"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,4) (0,3) (0,13) (1,6) (1,10) (1,4) (1,2) (2,3) (2,4) (2,10) (2,5) (2,9) (2,6) (3,12) (3,7) (4,7) (4,9) (4,16) (4,15) (5,8) (5,6) (5,10) (5,9) (5,14) (5,12) (5,11) (5,15) (6,15) (6,8) (6,14) (6,9) (6,7) (7,9) (7,11) (7,13) (7,8) (7,12) (8,9) (8,10) (8,15) (8,14) (8,16) (9,10) (9,11) (10,15) (10,12) (10,11) (10,13) (11,16) (11,12) (12,16) (12,13) (13,15) (14,16) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,13,15,14,16,12,11,10,9,8,7,4,1,6,5,2,3", "difficulty": "hard", "doc_id": "88"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,9) (0,17) (0,11) (0,1) (0,4) (0,12) (0,14) (0,19) (0,3) (0,5) (0,8) (0,13) (1,10) (1,13) (1,16) (1,17) (1,19) (1,2) (1,18) (1,11) (1,5) (1,3) (2,7) (2,10) (2,12) (2,15) (2,3) (2,19) (2,16) (2,8) (2,6) (2,18) (2,14) (2,17) (2,4) (3,19) (3,15) (3,17) (3,5) (3,6) (3,7) (3,4) (3,11) (3,18) (3,14) (3,12) (4,16) (4,13) (4,6) (4,19) (4,8) (4,5) (4,7) (4,14) (4,9) (4,11) (4,18) (4,10) (5,11) (5,18) (5,15) (5,9) (5,14) (5,12) (5,13) (5,6) (6,11) (6,12) (6,18) (6,14) (6,10) (6,9) (6,16) (6,19) (7,15) (7,9) (7,12) (7,14) (7,18) (7,8) (8,18) (8,13) (8,15) (8,10) (8,19) (8,17) (9,10) (9,16) (9,17) (9,11) (9,12) (9,18) (9,15) (9,14) (9,13) (10,13) (10,12) (10,15) (10,17) (10,16) (10,14) (10,11) (11,18) (11,12) (11,19) (11,13) (11,15) (12,14) (12,17) (12,19) (12,15) (13,14) (13,16) (13,15) (14,16) (14,15) (14,18) (15,17) (16,17) (16,19) (16,18) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,13,15,17,18,16,19,12,14,10,11,9,7,8,4,5,6,3,2,1", "difficulty": "hard", "doc_id": "89"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,6) (0,7) (0,2) (1,6) (1,10) (1,9) (2,5) (2,3) (2,8) (3,5) (3,9) (3,7) (3,4) (4,10) (4,9) (4,6) (5,8) (6,10) (6,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,5,8,9,6,1,10,4,3,7", "difficulty": "hard", "doc_id": "90"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (0,7) (0,4) (0,3) (1,4) (1,8) (1,7) (1,5) (1,3) (2,9) (2,7) (2,3) (2,6) (3,4) (3,8) (3,5) (3,9) (4,5) (4,6) (4,9) (5,8) (5,7) (6,9) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,9,8,7,5,1,4,6,2", "difficulty": "easy", "doc_id": "91"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,6) (0,14) (0,8) (0,4) (0,3) (0,12) (0,2) (0,13) (0,10) (0,5) (0,16) (0,15) (1,5) (1,3) (1,4) (1,8) (1,11) (1,6) (2,7) (2,5) (2,16) (2,12) (2,15) (2,8) (2,3) (2,6) (2,9) (2,4) (2,11) (3,6) (3,13) (3,15) (3,4) (3,14) (3,12) (4,7) (4,9) (4,6) (4,11) (4,15) (4,8) (4,16) (5,10) (5,6) (5,7) (5,15) (5,9) (6,15) (6,8) (6,12) (6,9) (7,10) (7,16) (7,9) (7,11) (8,10) (8,15) (8,16) (8,13) (9,11) (9,13) (9,16) (9,10) (9,12) (10,14) (10,11) (10,12) (10,13) (10,15) (10,16) (11,16) (11,13) (11,14) (12,13) (12,16) (12,15) (13,16) (13,15) (14,16) (14,15) (15,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,15,16,14,11,13,12,10,9,7,5,6,8,4,2,3,1", "difficulty": "hard", "doc_id": "92"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,3) (0,7) (0,4) (0,13) (0,2) (0,14) (1,10) (1,14) (1,9) (1,13) (2,10) (2,13) (2,6) (2,4) (3,5) (3,11) (3,12) (4,5) (4,7) (4,14) (5,6) (5,12) (5,11) (5,10) (5,13) (6,7) (6,13) (7,13) (7,11) (8,9) (8,14) (8,13) (8,10) (9,14) (9,13) (10,14) (11,12) (11,14) (11,13) (12,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,14,12,11,13,8,9,1,10,2,4,7,6,5,3", "difficulty": "hard", "doc_id": "93"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,4) (0,15) (0,1) (0,8) (0,5) (0,2) (0,14) (0,13) (0,3) (0,10) (1,10) (1,12) (1,3) (1,14) (1,4) (1,16) (1,6) (1,8) (1,5) (1,13) (1,11) (2,8) (2,5) (2,13) (2,9) (2,4) (2,15) (2,10) (2,7) (2,6) (3,15) (3,7) (3,16) (3,12) (3,13) (3,4) (3,5) (4,11) (4,5) (4,15) (4,10) (4,14) (4,13) (4,6) (4,16) (4,7) (4,9) (5,8) (5,15) (5,13) (5,11) (5,10) (5,12) (5,7) (5,14) (5,9) (6,13) (6,9) (6,11) (6,14) (6,12) (6,15) (6,10) (6,16) (7,14) (7,9) (7,13) (7,10) (7,15) (7,11) (8,16) (8,14) (8,13) (8,11) (8,9) (8,15) (9,11) (9,10) (9,16) (9,12) (9,13) (10,16) (11,12) (11,14) (11,16) (11,15) (12,15) (13,15) (13,16) (13,14) (15,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,16,15,13,14,11,12,9,8,5,7,4,3,1,6,2", "difficulty": "hard", "doc_id": "94"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,2) (1,5) (1,4) (1,3) (1,2) (2,6) (3,7) (4,7) (4,6) (5,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,6,4,7,5,1,3", "difficulty": "easy", "doc_id": "95"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,9) (0,14) (0,12) (0,8) (0,2) (0,4) (0,3) (0,11) (0,10) (0,13) (0,1) (1,11) (1,15) (1,7) (1,8) (1,4) (1,9) (2,8) (2,14) (2,5) (2,9) (2,12) (2,6) (3,10) (3,15) (3,12) (3,11) (3,6) (3,5) (3,7) (4,10) (4,15) (4,14) (4,7) (4,12) (4,11) (5,12) (5,14) (5,13) (5,6) (5,9) (5,10) (5,8) (5,7) (6,10) (6,14) (6,12) (6,11) (6,9) (6,7) (6,8) (7,8) (7,11) (7,12) (7,10) (7,15) (8,12) (8,10) (8,11) (9,15) (9,10) (9,12) (9,14) (9,13) (10,12) (10,15) (10,14) (10,11) (10,13) (11,13) (11,15) (11,14) (12,13) (12,15) (12,14) (13,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,9,13,15,12,14,11,10,8,6,2,5,3,7,4", "difficulty": "hard", "doc_id": "96"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,2) (0,7) (0,4) (1,4) (1,6) (1,3) (1,2) (3,4) (3,7) (3,5) (3,6) (4,7) (5,6) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,7,6,5,3,1,2", "difficulty": "easy", "doc_id": "97"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,3) (0,8) (0,9) (0,4) (0,7) (0,2) (0,10) (0,5) (1,2) (1,3) (1,6) (1,9) (2,6) (2,3) (2,9) (2,7) (3,4) (3,5) (3,9) (4,5) (4,8) (4,7) (5,10) (5,7) (5,6) (6,8) (6,9) (6,7) (7,9) (7,10) (8,9) (8,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,6,7,10,8,9,2,1,3,4", "difficulty": "hard", "doc_id": "98"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (0,3) (1,2) (1,3) (2,5) (2,8) (3,9) (3,4) (3,8) (3,6) (4,9) (4,6) (5,8) (5,6) (6,9) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,1,2,8,5,6,4,9,7", "difficulty": "easy", "doc_id": "99"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,5) (0,15) (0,2) (0,14) (0,17) (1,17) (1,7) (1,14) (1,9) (2,15) (2,4) (2,16) (2,12) (2,13) (2,10) (3,14) (3,8) (3,18) (3,12) (3,11) (3,17) (3,7) (3,10) (4,9) (4,17) (4,12) (4,14) (4,10) (4,8) (4,7) (5,15) (5,17) (5,9) (5,13) (5,12) (6,9) (6,12) (6,11) (6,14) (6,8) (6,7) (7,17) (7,11) (7,8) (7,10) (7,18) (8,10) (8,18) (8,11) (8,12) (8,15) (9,12) (10,12) (10,17) (10,15) (10,13) (11,18) (11,16) (12,17) (12,18) (12,15) (12,14) (14,17) (14,16) (14,15) (14,18) (15,18) (16,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,17,14,18,16,11,8,15,12,6,7,3,10,4,2,13,5,9,1", "difficulty": "hard", "doc_id": "100"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,11) (0,7) (0,4) (0,17) (1,6) (1,17) (1,19) (1,2) (1,18) (1,5) (1,16) (1,4) (1,14) (1,8) (2,16) (2,13) (2,15) (2,3) (2,19) (3,15) (3,14) (3,18) (3,4) (3,6) (4,9) (4,11) (4,15) (4,8) (4,6) (4,19) (5,19) (5,14) (5,8) (5,18) (5,7) (5,15) (5,17) (5,11) (6,13) (6,12) (6,15) (7,9) (7,16) (7,15) (7,12) (7,11) (7,19) (7,18) (8,10) (8,15) (8,14) (8,17) (8,16) (8,11) (9,19) (9,18) (9,15) (9,13) (10,15) (10,17) (11,12) (11,16) (11,19) (11,17) (12,16) (12,19) (12,18) (13,15) (13,14) (13,16) (14,19) (15,19) (15,17) (16,17) (16,19) (17,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,17,19,16,13,14,8,11,12,18,9,7,5,1,4,6,3,2,15,10", "difficulty": "hard", "doc_id": "101"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,5) (0,1) (0,8) (1,4) (1,9) (1,5) (1,3) (1,6) (2,7) (2,3) (2,5) (2,4) (2,9) (3,4) (3,7) (3,9) (3,5) (4,8) (4,9) (4,10) (6,10) (6,9) (6,7) (6,8) (7,9) (7,10) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,7,10,6,9,4,3,2,5,1", "difficulty": "hard", "doc_id": "102"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,15) (0,1) (0,8) (0,3) (0,4) (0,14) (0,2) (0,10) (0,9) (1,6) (1,11) (1,8) (1,3) (1,10) (1,5) (1,2) (2,7) (2,10) (2,4) (2,11) (2,5) (2,3) (2,6) (2,13) (3,8) (3,5) (3,4) (3,9) (3,13) (3,14) (4,8) (4,11) (4,13) (4,7) (4,15) (4,10) (5,8) (5,14) (5,7) (5,12) (6,7) (6,10) (6,15) (6,13) (6,12) (7,14) (7,10) (7,13) (7,9) (7,15) (8,15) (8,12) (8,10) (9,14) (9,11) (9,10) (9,15) (10,12) (10,13) (11,12) (12,14) (12,15) (12,13) (13,15) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,15,14,12,13,10,8,5,7,6,2,3,4,11,1", "difficulty": "hard", "doc_id": "103"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,10) (0,11) (0,1) (0,5) (0,15) (0,12) (0,2) (1,9) (1,2) (1,10) (1,5) (1,16) (2,6) (2,3) (2,14) (2,4) (2,12) (2,9) (2,8) (3,14) (3,7) (3,5) (3,6) (3,10) (4,13) (4,7) (4,9) (5,9) (5,6) (5,10) (5,8) (6,7) (6,11) (6,14) (6,12) (7,12) (7,9) (7,8) (7,10) (7,15) (8,13) (8,9) (9,13) (9,14) (9,11) (9,16) (9,17) (10,11) (10,16) (10,14) (10,15) (11,16) (11,12) (12,17) (12,14) (12,16) (13,14) (13,16) (14,15) (14,17) (15,16) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,8,9,17,16,15,14,13,4,7,10,11,12,6,3,5,1", "difficulty": "hard", "doc_id": "104"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (0,2) (1,7) (1,2) (2,8) (2,9) (3,8) (3,5) (3,7) (4,9) (4,7) (4,8) (6,8) (6,9) (7,8) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,1,7,4,9,6,8,3,5", "difficulty": "easy", "doc_id": "105"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,3) (0,10) (0,2) (0,5) (1,10) (1,3) (1,6) (1,5) (1,7) (2,7) (2,4) (2,6) (2,9) (2,10) (2,8) (3,10) (3,9) (3,6) (3,5) (3,7) (3,4) (4,10) (4,8) (4,7) (4,5) (5,6) (5,9) (5,10) (6,8) (6,7) (6,9) (7,8) (8,9) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,10,9,8,7,6,2,4,3,1", "difficulty": "hard", "doc_id": "106"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,6) (0,5) (0,2) (1,5) (1,7) (1,4) (1,6) (2,3) (2,6) (3,5) (3,7) (3,4) (4,6) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,6,5,3,4,1,7", "difficulty": "easy", "doc_id": "107"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,3) (0,4) (0,6) (0,5) (0,1) (0,7) (1,3) (1,8) (1,7) (1,4) (1,2) (2,8) (2,3) (2,6) (3,8) (3,7) (4,7) (4,8) (4,5) (5,6) (5,8) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,8,5,6,2,3,1,4", "difficulty": "easy", "doc_id": "108"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,11) (0,6) (1,7) (1,9) (1,5) (2,13) (2,6) (2,7) (2,11) (2,9) (3,12) (3,13) (3,4) (3,10) (3,8) (4,6) (4,10) (4,12) (4,9) (4,8) (5,13) (5,11) (5,7) (5,6) (6,10) (6,13) (6,11) (7,13) (7,10) (7,12) (7,9) (8,12) (8,10) (9,12) (10,11) (11,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,11,13,5,1,9,12,8,4,3,10,7,2", "difficulty": "hard", "doc_id": "109"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,5) (0,7) (0,4) (0,2) (1,4) (1,3) (2,4) (2,5) (3,6) (3,7) (4,7) (4,5) (5,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,5,7,4,1,3,6", "difficulty": "easy", "doc_id": "110"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,1) (0,15) (0,8) (0,14) (0,18) (0,3) (0,6) (0,12) (0,16) (0,10) (0,7) (0,17) (1,17) (1,18) (1,5) (1,16) (1,10) (1,9) (1,15) (1,14) (1,13) (1,8) (1,2) (2,6) (2,8) (2,16) (2,12) (2,14) (2,15) (2,18) (2,9) (2,13) (2,5) (2,11) (3,7) (3,4) (3,13) (3,18) (3,10) (3,8) (3,12) (3,15) (3,11) (4,6) (4,13) (4,16) (4,8) (4,5) (4,14) (4,18) (4,10) (4,7) (4,17) (5,8) (5,10) (5,6) (5,16) (5,17) (5,18) (5,14) (5,15) (6,12) (6,16) (6,14) (6,9) (7,10) (7,12) (7,11) (7,16) (8,11) (8,16) (8,10) (8,12) (8,9) (8,18) (8,17) (9,11) (9,17) (9,18) (9,14) (9,16) (9,15) (10,15) (10,18) (11,17) (11,12) (11,16) (11,18) (11,14) (12,14) (12,18) (12,16) (12,13) (13,18) (13,17) (13,16) (14,16) (14,17) (14,15) (15,17) (15,18) (16,18) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,17,18,16,14,15,10,8,9,11,12,13,3,7,4,5,6,2,1", "difficulty": "hard", "doc_id": "111"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,10) (0,3) (1,4) (1,5) (1,3) (1,8) (2,7) (2,8) (2,3) (2,5) (3,7) (3,6) (4,8) (5,8) (5,7) (6,8) (7,10) (7,9) (8,9) (8,10) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,6,8,10,9,7,2,5,1,4", "difficulty": "hard", "doc_id": "112"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,9) (0,18) (0,6) (0,15) (0,16) (0,17) (0,3) (0,19) (0,4) (0,5) (0,12) (1,15) (1,10) (1,6) (1,19) (1,14) (2,18) (2,17) (2,3) (2,8) (2,14) (2,10) (3,9) (3,8) (3,14) (3,12) (3,10) (4,8) (4,7) (4,17) (4,16) (4,5) (4,19) (5,19) (5,8) (5,18) (5,17) (5,15) (5,9) (6,10) (6,8) (6,11) (6,14) (6,13) (7,8) (7,18) (7,14) (8,13) (8,11) (8,17) (8,19) (8,10) (9,17) (9,14) (9,10) (9,18) (9,13) (10,18) (10,11) (10,14) (10,16) (10,13) (11,13) (11,16) (12,15) (12,16) (13,19) (13,15) (14,18) (15,16) (16,18) (16,19) (16,17) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,16,17,18,14,10,13,15,5,9,3,2,8,11,6,1,19,4,7", "difficulty": "hard", "doc_id": "113"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,2) (0,3) (1,3) (1,2) (1,6) (2,7) (2,3) (2,4) (2,6) (2,5) (4,7) (5,7) (5,6) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,2,4,7,5,6,1", "difficulty": "easy", "doc_id": "114"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,2) (0,1) (0,6) (0,8) (0,4) (1,4) (1,7) (2,6) (2,8) (2,9) (3,9) (3,7) (3,10) (3,8) (4,9) (4,7) (4,10) (5,8) (5,10) (5,7) (5,6) (6,8) (6,9) (6,7) (7,9) (8,10) (8,9) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,10,9,3,8,2,6,5,7,1", "difficulty": "hard", "doc_id": "115"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,3) (0,1) (0,4) (0,2) (1,5) (2,4) (2,5) (2,3) (2,6) (3,4) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,3,2,6,5,1", "difficulty": "easy", "doc_id": "116"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,8) (0,2) (0,7) (0,1) (0,6) (1,3) (1,6) (1,7) (1,8) (1,4) (2,3) (2,7) (2,6) (2,8) (3,4) (3,6) (4,7) (4,6) (5,8) (6,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,4,7,2,3,1,8,5", "difficulty": "easy", "doc_id": "117"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,4) (0,11) (0,10) (0,13) (1,4) (1,7) (1,13) (1,11) (1,10) (2,7) (2,11) (2,8) (2,4) (3,11) (3,4) (3,7) (4,11) (4,8) (5,12) (5,10) (5,9) (5,8) (5,7) (6,13) (6,11) (7,9) (7,11) (7,8) (8,10) (9,11) (9,13) (10,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,13,6,11,3,4,2,8,10,1,7,9,5,12", "difficulty": "hard", "doc_id": "118"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (1,4) (2,4) (2,5) (3,4) (3,5) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,4,3,5,2", "difficulty": "easy", "doc_id": "119"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,16) (0,12) (0,7) (0,8) (0,9) (0,13) (0,3) (0,6) (0,11) (0,19) (0,2) (0,1) (1,5) (1,8) (1,11) (1,3) (1,4) (1,14) (1,9) (1,19) (1,2) (1,15) (1,7) (1,16) (2,6) (2,15) (2,9) (2,11) (2,5) (2,19) (2,3) (2,8) (2,10) (2,13) (2,18) (3,18) (3,5) (3,16) (3,7) (3,17) (3,6) (3,12) (3,14) (3,19) (3,8) (4,11) (4,8) (4,19) (4,9) (4,18) (4,17) (4,15) (4,14) (4,16) (4,10) (5,10) (5,6) (5,14) (5,16) (5,15) (5,12) (5,18) (6,11) (6,15) (6,13) (6,9) (6,7) (6,18) (6,19) (6,12) (6,14) (7,16) (7,15) (7,14) (7,10) (7,12) (7,19) (7,18) (7,9) (8,19) (8,12) (8,15) (8,17) (8,14) (8,13) (8,16) (9,12) (9,10) (9,15) (9,11) (9,19) (9,16) (9,18) (9,17) (10,14) (10,11) (10,12) (10,13) (10,16) (10,18) (11,19) (11,17) (11,12) (11,13) (12,15) (12,13) (12,14) (12,17) (12,16) (13,19) (13,16) (13,15) (14,18) (14,16) (15,19) (15,18) (15,17) (16,17) (17,19) (17,18) (18,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,16,17,18,19,15,13,12,14,10,11,9,7,6,5,3,2,8,4", "difficulty": "hard", "doc_id": "120"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,5) (1,3) (1,5) (1,2) (1,4) (2,5) (3,4)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,2,1,4,3", "difficulty": "easy", "doc_id": "121"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,3) (0,4) (0,2) (0,5) (1,2) (1,5) (1,7) (1,3) (2,4) (2,7) (2,5) (3,4) (3,6) (3,5) (4,6) (4,7) (4,5) (5,7) (5,6) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,6,7,4,3,1,2", "difficulty": "easy", "doc_id": "122"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,4) (1,5) (1,3) (1,2) (1,4) (2,4) (3,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,2,1,3,5", "difficulty": "easy", "doc_id": "123"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,4) (0,5) (1,10) (1,13) (1,2) (1,8) (2,3) (2,9) (3,13) (3,5) (3,14) (3,10) (3,7) (4,13) (4,8) (4,5) (4,12) (5,12) (5,6) (5,13) (5,10) (6,12) (6,7) (7,12) (7,9) (7,10) (7,14) (8,10) (8,11) (8,12) (8,13) (8,14) (9,13) (9,14) (9,12) (10,14) (11,12) (11,13) (12,14) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,10,14,9,12,11,13,4,8,1,2,3,7,6", "difficulty": "hard", "doc_id": "124"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,4) (0,8) (1,7) (2,7) (2,4) (2,6) (2,3) (3,8) (3,5) (3,6) (3,4) (4,7) (4,9) (4,5) (5,7) (5,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,9,5,4,3,6,2,7,1", "difficulty": "easy", "doc_id": "125"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,8) (0,3) (0,7) (0,1) (1,4) (1,3) (1,2) (1,7) (2,3) (2,8) (2,9) (2,6) (3,8) (3,7) (3,6) (3,4) (3,5) (4,8) (4,5) (4,9) (5,9) (5,6) (5,8) (6,9) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,7,9,8,5,6,2,3,4", "difficulty": "easy", "doc_id": "126"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,2) (0,9) (1,8) (1,6) (2,4) (2,5) (2,9) (3,9) (3,6) (3,4) (4,9) (4,8) (5,9) (5,7) (5,6) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,5,7,8,1,6,3,4,2", "difficulty": "easy", "doc_id": "127"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,4) (0,8) (0,3) (1,2) (1,8) (1,5) (1,4) (1,6) (2,6) (2,4) (3,4) (3,6) (3,7) (4,7) (4,6) (5,7) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,7,8,6,4,2,1,5", "difficulty": "easy", "doc_id": "128"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,13) (0,8) (0,5) (0,15) (0,16) (0,1) (1,2) (1,3) (1,8) (1,15) (1,4) (1,10) (1,16) (1,7) (1,13) (2,4) (2,13) (2,8) (2,14) (2,6) (2,3) (3,9) (3,4) (3,13) (3,15) (3,12) (3,7) (4,10) (4,16) (5,6) (5,16) (5,15) (5,14) (5,7) (5,11) (6,11) (6,14) (6,10) (7,10) (7,16) (7,14) (7,8) (8,14) (8,9) (9,15) (10,15) (10,11) (12,13) (12,16) (13,14) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,13,14,16,12,3,7,8,9,15,10,11,5,6,2,4", "difficulty": "hard", "doc_id": "129"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,6) (0,5) (0,9) (0,11) (0,14) (0,4) (0,8) (0,12) (0,10) (0,7) (1,17) (1,10) (2,17) (2,3) (2,8) (2,4) (2,10) (2,12) (3,6) (3,14) (3,9) (3,15) (3,13) (3,18) (3,16) (4,10) (4,17) (4,5) (4,9) (4,15) (4,18) (5,13) (5,12) (5,16) (5,7) (5,10) (5,8) (5,17) (6,16) (6,17) (6,14) (6,15) (6,10) (6,11) (6,18) (7,18) (7,8) (7,10) (7,15) (8,15) (8,17) (8,13) (8,9) (8,18) (9,10) (9,14) (9,11) (9,17) (9,12) (9,16) (9,18) (10,11) (10,12) (10,17) (10,15) (11,14) (12,13) (12,18) (13,17) (13,16) (13,14) (13,15) (14,17) (16,17) (16,18) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,15,13,14,17,18,16,3,6,11,9,12,5,8,2,4,10,1", "difficulty": "hard", "doc_id": "130"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,7) (0,2) (0,8) (0,1) (0,4) (1,5) (1,2) (1,7) (1,8) (1,6) (2,8) (2,3) (3,4) (3,8) (4,5) (4,6) (4,8) (4,7) (5,8) (5,6) (5,7) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,7,8,6,5,1,2,3", "difficulty": "easy", "doc_id": "131"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,8) (0,2) (1,2) (1,7) (1,5) (2,3) (2,4) (2,8) (2,7) (3,4) (4,8) (4,7) (5,8) (5,6) (6,8) (6,9) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,3,4,7,1,5,8,6,9", "difficulty": "easy", "doc_id": "132"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,3) (0,2) (0,6) (0,7) (1,2) (1,4) (1,6) (1,3) (2,7) (2,6) (2,3) (2,8) (2,5) (2,4) (3,4) (3,5) (3,8) (3,6) (4,6) (4,5) (5,8) (5,7) (6,7) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,8,6,4,5,3,2,1", "difficulty": "easy", "doc_id": "133"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (0,7) (0,4) (0,5) (0,2) (1,2) (1,3) (1,8) (1,4) (1,5) (1,6) (2,5) (2,9) (2,7) (2,4) (2,6) (3,8) (3,6) (3,9) (3,4) (4,5) (4,9) (5,8) (5,7) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,6,3,4,9,8,1,5,7", "difficulty": "easy", "doc_id": "134"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,6) (0,4) (0,7) (0,5) (0,1) (1,4) (1,7) (2,7) (2,3) (3,5) (3,4) (3,6) (3,7) (4,7) (4,6) (5,6) (5,7) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,7,5,6,4,3,2", "difficulty": "easy", "doc_id": "135"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,1) (0,5) (0,2) (1,3) (1,5) (1,6) (1,4) (1,2) (2,4) (2,5) (3,5) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,4,5,3,1,6", "difficulty": "easy", "doc_id": "136"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,3) (0,6) (0,10) (0,7) (0,4) (1,5) (1,3) (1,6) (1,2) (1,9) (1,8) (1,4) (2,9) (2,3) (2,5) (2,4) (2,11) (2,7) (3,10) (3,7) (3,11) (4,6) (5,6) (6,8) (6,11) (7,11) (8,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,6,11,7,3,10,8,1,9,2,5", "difficulty": "hard", "doc_id": "137"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,5) (0,13) (0,11) (0,7) (0,3) (0,6) (0,1) (0,8) (1,10) (1,9) (1,11) (1,14) (2,3) (2,11) (2,7) (2,13) (2,14) (2,12) (3,8) (3,14) (3,6) (3,7) (4,14) (4,10) (4,8) (4,6) (4,5) (4,9) (5,10) (5,12) (5,8) (5,14) (5,11) (6,9) (6,13) (6,14) (7,12) (7,9) (7,13) (7,11) (7,10) (8,14) (9,10) (10,13) (11,12) (11,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,14,6,13,11,12,5,4,9,1,10,7,3,2", "difficulty": "hard", "doc_id": "138"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,13) (0,15) (0,1) (0,4) (0,16) (0,2) (1,5) (1,14) (1,12) (1,4) (2,9) (2,15) (2,11) (2,5) (2,6) (2,17) (3,17) (3,6) (3,7) (3,5) (3,16) (3,10) (3,11) (4,10) (4,12) (4,11) (4,9) (5,13) (5,12) (5,14) (5,10) (5,8) (6,17) (6,14) (6,13) (6,7) (6,9) (6,10) (6,8) (7,15) (7,12) (7,16) (8,15) (8,17) (8,12) (8,13) (8,11) (9,11) (9,12) (9,10) (10,13) (10,17) (10,12) (11,15) (11,17) (12,16) (12,14) (13,14) (13,17) (13,15) (13,16) (14,15) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,17,13,16,14,15,11,9,10,12,8,6,7,3,5,1,4", "difficulty": "hard", "doc_id": "139"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,3) (0,2) (0,7) (1,5) (1,3) (1,8) (2,8) (2,6) (4,8) (4,7) (4,6) (4,5) (5,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,5,4,6,2,8,1,3", "difficulty": "easy", "doc_id": "140"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,2) (0,5) (0,6) (0,4) (1,3) (1,4) (1,2) (1,6) (1,8) (2,3) (2,6) (2,4) (3,5) (3,6) (3,8) (6,8) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,2,6,7,8,1,3,5", "difficulty": "easy", "doc_id": "141"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,8) (0,10) (0,7) (0,6) (0,4) (0,1) (0,13) (0,11) (1,12) (1,14) (1,3) (1,13) (1,8) (2,5) (2,4) (2,14) (3,12) (3,5) (3,7) (3,9) (4,5) (4,9) (4,6) (5,7) (5,8) (5,14) (5,13) (6,8) (6,7) (7,10) (7,9) (7,12) (7,8) (8,9) (8,10) (8,11) (9,12) (9,11) (10,13) (10,14) (10,12) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,12,10,14,5,13,1,8,9,3,7,6,4,2", "difficulty": "hard", "doc_id": "142"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,9) (0,2) (0,1) (0,10) (0,4) (0,6) (0,11) (0,7) (1,3) (1,2) (1,4) (1,11) (2,3) (2,8) (2,7) (2,5) (2,10) (3,4) (3,5) (4,9) (5,6) (5,10) (6,8) (7,11) (7,9) (7,10) (8,11) (9,10) (9,11) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,10,11,9,4,3,5,6,8,2,1", "difficulty": "hard", "doc_id": "143"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,6) (0,4) (0,9) (0,7) (0,1) (1,2) (1,4) (1,9) (1,8) (1,5) (2,8) (2,5) (3,4) (3,8) (3,5) (3,6) (4,9) (4,7) (5,8) (6,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,2,5,3,6,8,9,4,7", "difficulty": "easy", "doc_id": "144"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,9) (0,7) (0,5) (0,1) (0,2) (0,6) (1,3) (1,5) (2,7) (2,4) (2,5) (2,8) (2,3) (2,11) (2,10) (3,6) (3,11) (3,4) (4,6) (4,5) (4,11) (5,6) (6,8) (6,9) (6,7) (7,10) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,5,4,3,11,10,7,2,8,6,9", "difficulty": "hard", "doc_id": "145"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,11) (0,17) (0,2) (0,16) (0,7) (0,6) (0,13) (0,1) (0,19) (0,5) (0,4) (1,5) (1,6) (1,16) (1,8) (1,7) (1,19) (1,14) (1,4) (2,6) (2,9) (2,18) (2,5) (2,15) (3,5) (3,14) (3,7) (3,9) (3,16) (3,13) (3,18) (4,7) (4,16) (4,19) (4,12) (4,14) (4,8) (4,5) (5,7) (5,11) (5,17) (5,16) (5,19) (6,15) (6,12) (6,11) (6,7) (7,9) (7,12) (7,17) (7,8) (7,16) (7,11) (7,19) (8,9) (8,11) (8,12) (8,13) (9,19) (9,18) (9,13) (9,10) (10,18) (10,17) (11,13) (11,14) (11,15) (11,12) (12,14) (12,18) (13,14) (13,17) (14,19) (14,18) (15,18) (15,19) (15,16) (15,17) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,5,19,15,17,18,14,13,11,12,8,7,3,16,1,6,2,9,10", "difficulty": "hard", "doc_id": "146"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,5) (0,17) (0,16) (0,15) (0,10) (0,7) (0,12) (0,3) (1,15) (1,6) (1,3) (1,10) (1,17) (1,5) (1,16) (1,11) (1,8) (1,12) (2,3) (2,15) (2,5) (2,17) (2,4) (2,7) (2,12) (2,10) (2,11) (2,16) (3,10) (3,8) (3,7) (3,6) (3,11) (3,13) (3,16) (3,4) (3,5) (3,14) (4,9) (4,15) (4,14) (4,5) (4,7) (4,17) (4,10) (5,11) (5,12) (5,15) (5,8) (5,10) (5,17) (5,6) (6,13) (6,16) (6,11) (6,14) (6,15) (6,12) (7,13) (7,9) (7,17) (7,15) (8,11) (8,14) (8,12) (8,10) (8,13) (8,17) (8,15) (8,9) (9,17) (9,14) (9,12) (9,16) (10,12) (10,11) (10,17) (10,14) (11,12) (11,16) (11,13) (11,15) (11,14) (12,15) (12,13) (12,14) (12,17) (13,14) (13,15) (13,16) (14,15) (15,17) (15,16) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,14,15,16,17,12,13,11,10,8,9,7,4,2,5,6,1", "difficulty": "hard", "doc_id": "147"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,7) (0,2) (0,1) (0,6) (1,7) (1,2) (2,3) (2,7) (3,6) (3,4) (4,5) (4,7) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,7,1,2,3,4,5", "difficulty": "easy", "doc_id": "148"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,6) (0,15) (0,7) (0,4) (1,15) (1,6) (2,7) (2,4) (2,12) (3,15) (3,12) (3,13) (4,15) (4,7) (4,9) (4,12) (4,6) (4,14) (5,15) (5,9) (5,12) (5,13) (6,7) (6,15) (6,11) (6,12) (7,8) (7,13) (7,9) (8,12) (8,15) (8,13) (9,11) (10,11) (10,14) (12,13) (12,15) (13,15) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,14,10,11,9,5,13,8,15,3,12,2,7,6,1", "difficulty": "hard", "doc_id": "149"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,5) (0,6) (0,2) (0,1) (1,6) (1,4) (1,3) (1,7) (1,8) (2,8) (2,5) (2,3) (2,4) (3,4) (3,7) (3,5) (4,5) (4,6) (4,8) (5,7) (6,8) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,8,7,6,4,5,3,2", "difficulty": "easy", "doc_id": "150"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,6) (0,9) (0,8) (0,7) (0,4) (1,5) (1,2) (1,7) (1,3) (1,10) (3,10) (3,6) (3,4) (3,7) (3,9) (3,8) (4,10) (4,5) (4,6) (4,7) (4,9) (5,10) (5,6) (6,7) (7,10) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,9,3,8,7,6,5,10,1,2", "difficulty": "hard", "doc_id": "151"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (0,4) (0,7) (0,6) (0,2) (1,8) (1,5) (2,6) (2,9) (2,4) (3,8) (3,9) (4,6) (4,8) (5,9) (5,8) (6,9) (6,7) (7,8) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,4,6,7,9,5,1,8,3", "difficulty": "easy", "doc_id": "152"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (0,5) (0,8) (1,7) (1,2) (1,5) (1,9) (2,4) (2,9) (3,6) (3,9) (3,8) (3,4) (4,5) (4,6) (4,9) (5,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,3,6,4,5,9,2,1,7", "difficulty": "easy", "doc_id": "153"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,5) (0,6) (0,7) (0,1) (0,11) (0,9) (0,13) (0,14) (0,4) (1,2) (1,8) (2,12) (2,10) (2,11) (2,4) (3,10) (3,11) (3,16) (4,5) (4,18) (4,15) (4,7) (4,14) (4,16) (4,17) (4,9) (5,9) (5,16) (5,18) (5,12) (6,14) (6,10) (6,9) (6,8) (6,11) (6,16) (7,18) (7,10) (7,9) (7,16) (8,16) (8,17) (8,10) (8,12) (9,18) (9,15) (10,13) (10,12) (10,16) (11,16) (11,18) (11,14) (11,13) (12,15) (12,17) (12,14) (13,15) (13,14) (14,16) (16,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,9,15,13,14,16,7,18,5,12,17,8,6,11,3,10,2,1", "difficulty": "hard", "doc_id": "154"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,10) (0,4) (0,11) (0,14) (0,1) (0,9) (0,2) (0,18) (0,7) (0,12) (1,17) (1,9) (1,16) (1,13) (1,6) (1,5) (1,10) (1,3) (1,12) (1,7) (1,18) (1,11) (2,6) (2,7) (2,15) (2,14) (2,13) (2,17) (2,5) (2,18) (3,14) (3,12) (3,18) (3,8) (3,4) (3,10) (3,7) (3,15) (4,15) (4,9) (4,5) (4,12) (4,13) (4,7) (4,6) (4,8) (4,14) (5,18) (5,13) (5,17) (5,11) (5,6) (5,14) (5,15) (5,10) (6,17) (6,13) (6,18) (6,15) (6,14) (6,9) (6,7) (6,16) (7,9) (7,15) (7,10) (7,16) (7,17) (7,8) (7,11) (7,12) (8,17) (8,18) (8,9) (8,11) (8,14) (9,18) (9,12) (9,16) (9,13) (9,15) (10,15) (10,14) (10,17) (10,18) (10,16) (10,13) (11,14) (11,13) (11,15) (11,12) (12,17) (12,16) (12,13) (12,15) (13,15) (13,16) (14,15) (14,17) (14,18) (15,16) (16,18) (16,17) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,15,16,17,18,14,11,13,10,7,8,9,6,4,3,1,5,2", "difficulty": "hard", "doc_id": "155"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,3) (0,4) (0,14) (0,7) (0,1) (0,8) (0,13) (0,6) (0,11) (0,12) (0,2) (0,9) (1,3) (1,9) (1,8) (1,10) (1,12) (1,7) (1,6) (1,14) (1,5) (1,13) (2,9) (2,11) (2,12) (2,6) (2,4) (2,3) (2,7) (2,14) (3,4) (3,12) (3,8) (3,9) (3,10) (4,11) (4,10) (4,6) (4,5) (4,14) (5,12) (5,6) (5,14) (5,8) (5,7) (6,10) (6,8) (6,7) (6,9) (6,11) (6,12) (6,13) (7,14) (7,12) (7,10) (7,8) (8,13) (8,9) (8,10) (8,14) (8,11) (9,10) (9,13) (11,12) (11,13) (12,14) (12,13) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,13,14,12,11,8,10,7,6,5,4,2,3,1", "difficulty": "hard", "doc_id": "156"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,2) (1,6) (1,4) (1,7) (1,3) (1,5) (2,6) (2,3) (3,6) (3,8) (3,7) (3,5) (4,6) (4,7) (5,8) (5,6) (6,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,3,5,8,6,4,7,1", "difficulty": "easy", "doc_id": "157"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,4) (0,7) (0,1) (0,6) (0,5) (0,3) (1,2) (1,5) (1,8) (1,7) (1,3) (1,6) (1,4) (2,3) (2,7) (2,8) (2,4) (3,5) (3,8) (3,4) (3,7) (4,8) (4,6) (5,8) (5,7) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,7,6,4,8,5,1,2", "difficulty": "easy", "doc_id": "158"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,2) (0,9) (0,12) (0,6) (0,5) (0,10) (1,13) (1,3) (1,2) (1,16) (2,15) (2,8) (2,3) (2,12) (2,16) (2,4) (2,7) (3,11) (3,9) (3,8) (3,7) (3,6) (4,13) (4,9) (5,13) (5,9) (5,8) (6,11) (6,10) (6,8) (6,13) (6,14) (6,7) (7,12) (7,9) (7,13) (7,15) (8,12) (8,14) (8,9) (8,15) (8,10) (9,16) (9,13) (9,15) (9,12) (9,14) (10,14) (10,15) (10,11) (10,13) (11,15) (11,14) (12,14) (13,14) (13,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,13,16,9,5,8,15,11,14,12,7,6,3,1,2,4", "difficulty": "hard", "doc_id": "159"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,11) (0,12) (0,8) (0,10) (0,4) (1,8) (1,7) (1,9) (1,10) (1,6) (1,3) (1,12) (1,11) (1,5) (2,4) (2,10) (2,9) (2,11) (2,5) (2,8) (2,12) (2,6) (2,7) (3,5) (3,7) (3,10) (3,12) (3,6) (4,12) (4,5) (4,6) (4,8) (5,7) (5,9) (5,12) (5,8) (5,6) (5,11) (5,10) (6,12) (6,7) (6,11) (6,8) (7,12) (7,10) (7,9) (7,8) (8,10) (8,11) (8,9) (9,10) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,8,9,10,11,6,7,12,3,1,5,2", "difficulty": "hard", "doc_id": "160"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,4) (0,9) (0,2) (0,6) (1,7) (1,8) (1,5) (1,2) (1,9) (1,4) (1,6) (2,8) (2,5) (2,9) (2,7) (3,7) (3,6) (4,7) (4,5) (4,8) (5,7) (5,9) (6,9) (6,8) (7,9) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,8,9,5,4,1,2,7,3", "difficulty": "easy", "doc_id": "161"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,13) (0,7) (0,1) (0,9) (0,4) (0,12) (0,6) (1,13) (1,8) (1,11) (1,10) (1,4) (1,5) (1,9) (1,6) (2,11) (2,9) (2,13) (2,3) (2,7) (2,14) (2,5) (2,4) (2,6) (3,4) (3,14) (3,13) (3,5) (3,10) (3,7) (3,8) (4,12) (4,7) (4,13) (4,6) (4,11) (4,8) (4,5) (4,10) (5,7) (5,6) (5,9) (5,11) (5,8) (5,10) (6,10) (6,9) (6,14) (7,12) (7,8) (7,14) (8,11) (8,10) (8,12) (8,9) (9,14) (9,13) (9,10) (10,11) (10,12) (10,13) (10,14) (11,13) (11,14) (12,13) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,14,13,12,10,11,8,9,5,7,3,2,4,1", "difficulty": "hard", "doc_id": "162"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,7) (0,8) (0,1) (0,5) (0,4) (1,5) (2,6) (2,5) (2,7) (2,8) (3,4) (3,8) (4,6) (4,5) (5,8) (6,8) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,3,8,7,6,2,5,1", "difficulty": "easy", "doc_id": "163"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,15) (0,1) (0,3) (0,6) (0,10) (0,7) (0,9) (0,17) (0,13) (0,8) (0,18) (1,11) (1,16) (1,5) (1,12) (1,9) (1,6) (1,17) (1,2) (2,15) (2,7) (2,3) (2,11) (2,12) (2,10) (3,7) (3,10) (3,8) (3,12) (3,11) (3,5) (3,16) (4,16) (4,15) (4,10) (5,15) (5,7) (5,11) (6,10) (6,15) (6,7) (6,9) (6,8) (6,17) (7,11) (7,14) (7,18) (7,9) (8,12) (8,17) (8,15) (9,17) (9,15) (10,13) (10,16) (10,18) (10,14) (11,15) (12,15) (13,17) (13,16) (13,18) (14,17) (14,15) (15,18) (15,16) (16,17) (16,18) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,18,17,16,13,10,14,7,9,6,8,12,3,5,11,1,2,15,4", "difficulty": "hard", "doc_id": "164"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,7) (0,1) (0,8) (0,6) (0,3) (0,9) (1,6) (1,4) (1,5) (1,7) (2,5) (2,3) (2,6) (2,9) (3,7) (3,8) (3,6) (3,4) (3,5) (4,8) (4,7) (4,6) (5,6) (5,8) (6,9) (6,8) (6,7) (7,8) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,8,7,6,5,2,3,4,1", "difficulty": "easy", "doc_id": "165"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,6) (0,3) (0,2) (0,7) (1,9) (2,6) (2,3) (2,5) (2,7) (3,7) (4,7) (4,8) (5,9) (5,8) (6,9) (7,8) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,7,4,8,5,2,6,9,1", "difficulty": "easy", "doc_id": "166"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,11) (0,5) (0,2) (0,6) (0,8) (0,10) (0,4) (0,7) (1,3) (1,5) (1,9) (1,7) (1,8) (1,10) (2,6) (2,4) (2,5) (2,11) (2,7) (3,8) (3,4) (3,10) (4,10) (4,9) (4,5) (4,8) (4,11) (4,7) (5,7) (5,10) (5,8) (5,6) (6,9) (6,7) (6,8) (6,10) (7,11) (7,9) (7,8) (8,11) (8,9) (8,10) (9,11) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,8,10,11,9,6,5,2,4,3,1", "difficulty": "hard", "doc_id": "167"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,2) (0,11) (0,18) (0,6) (0,7) (0,14) (0,1) (0,19) (0,15) (0,13) (1,7) (1,8) (1,2) (1,16) (1,13) (1,12) (1,18) (1,4) (1,19) (1,14) (1,9) (1,5) (1,3) (2,4) (2,14) (2,5) (2,8) (2,7) (2,12) (2,15) (2,17) (2,13) (2,11) (3,15) (3,19) (3,16) (3,13) (3,7) (3,11) (3,10) (3,6) (3,17) (4,12) (4,6) (4,15) (4,9) (4,5) (4,16) (4,11) (4,18) (5,19) (5,15) (5,13) (5,9) (5,18) (5,12) (5,16) (5,11) (6,13) (6,9) (6,14) (6,7) (6,8) (6,19) (7,11) (7,9) (7,17) (7,16) (7,12) (7,13) (8,10) (8,15) (8,11) (8,12) (8,16) (8,18) (8,19) (9,14) (9,16) (9,19) (9,10) (9,18) (9,15) (9,12) (10,19) (10,15) (10,14) (10,16) (10,13) (10,17) (10,12) (11,19) (11,12) (11,15) (12,15) (12,16) (12,14) (12,19) (13,14) (13,18) (13,16) (13,15) (14,16) (14,19) (14,17) (14,15) (15,19) (15,18) (16,17) (16,18) (18,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,13,15,18,19,14,17,16,12,11,8,10,9,7,6,4,5,2,1,3", "difficulty": "hard", "doc_id": "168"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,8) (0,10) (0,2) (0,7) (1,6) (1,7) (1,11) (1,5) (2,3) (3,10) (3,11) (3,9) (4,8) (4,6) (5,12) (5,7) (5,10) (5,11) (5,9) (6,11) (6,12) (7,12) (8,11) (8,12) (8,10) (9,10) (9,12) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,12,11,8,4,6,1,5,9,10,3,2", "difficulty": "hard", "doc_id": "169"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,7) (0,1) (0,3) (0,8) (1,2) (1,3) (1,7) (1,6) (2,9) (2,5) (2,7) (2,6) (2,4) (2,8) (3,8) (3,6) (3,9) (4,6) (4,8) (4,7) (4,9) (5,6) (5,7) (6,8) (6,9) (7,9) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,7,9,4,6,5,2,1,3", "difficulty": "easy", "doc_id": "170"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,4) (0,6) (0,13) (0,2) (0,5) (0,10) (1,13) (1,9) (1,11) (1,7) (1,5) (1,4) (1,14) (2,7) (2,6) (3,11) (3,7) (3,6) (3,9) (3,5) (4,12) (4,8) (5,7) (5,6) (5,8) (5,14) (5,13) (5,12) (5,10) (6,10) (6,11) (6,9) (6,13) (7,12) (7,11) (7,15) (7,10) (7,8) (8,10) (8,14) (8,11) (8,15) (9,12) (9,10) (9,14) (10,14) (10,13) (11,12) (13,15) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,13,14,9,12,11,8,15,7,2,6,3,5,1,4", "difficulty": "hard", "doc_id": "171"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,9) (0,1) (0,5) (0,6) (0,16) (0,10) (1,6) (1,11) (2,15) (2,3) (2,11) (2,9) (2,5) (2,16) (3,15) (3,14) (3,6) (3,7) (4,11) (4,7) (4,16) (4,6) (4,17) (5,15) (5,11) (5,13) (6,16) (6,9) (6,7) (6,8) (6,14) (6,13) (7,8) (7,13) (7,10) (8,9) (8,13) (8,10) (8,16) (9,16) (9,15) (9,11) (9,13) (10,14) (10,16) (11,17) (11,12) (11,13) (11,14) (12,15) (12,14) (12,13) (12,17) (14,16) (15,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,16,14,12,17,15,9,13,8,7,4,6,3,2,5,11,1", "difficulty": "hard", "doc_id": "172"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (0,6) (0,4) (1,2) (1,4) (1,6) (2,3) (2,4) (2,5) (2,9) (3,6) (3,7) (3,4) (4,6) (5,8) (5,9) (6,8) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,6,3,7,8,5,9,2,1", "difficulty": "easy", "doc_id": "173"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,2) (0,13) (0,15) (0,9) (0,10) (0,6) (0,4) (0,7) (0,18) (0,17) (0,5) (0,12) (0,1) (1,11) (1,9) (1,18) (1,10) (1,2) (1,3) (1,6) (1,17) (1,14) (1,16) (1,12) (2,16) (2,6) (2,12) (2,8) (2,11) (2,17) (2,9) (2,7) (2,13) (2,18) (3,14) (3,11) (3,12) (3,15) (3,18) (3,7) (3,19) (3,4) (3,10) (3,6) (3,5) (3,16) (4,7) (4,11) (4,12) (4,8) (4,6) (4,9) (4,14) (4,13) (5,12) (5,18) (5,19) (5,14) (5,8) (5,7) (5,6) (5,17) (6,12) (6,7) (6,16) (6,8) (6,13) (7,11) (7,15) (7,9) (7,16) (7,18) (7,8) (7,12) (8,12) (8,14) (8,16) (8,18) (8,11) (8,17) (8,9) (8,10) (9,10) (9,17) (9,19) (9,14) (9,18) (9,16) (10,11) (10,19) (10,14) (10,16) (10,12) (10,18) (10,17) (11,16) (11,12) (12,17) (12,14) (12,18) (12,15) (12,13) (13,18) (13,14) (13,19) (14,16) (14,15) (14,17) (15,19) (15,17) (15,16) (16,17) (16,18) (16,19) (18,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,12,13,19,18,16,17,15,14,10,11,8,9,7,5,3,4,6,2", "difficulty": "hard", "doc_id": "174"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,10) (0,6) (0,5) (1,3) (1,6) (1,2) (1,7) (1,5) (1,10) (1,8) (1,9) (2,4) (2,7) (2,9) (2,5) (2,6) (3,11) (3,5) (3,4) (3,6) (4,5) (4,10) (5,10) (5,6) (5,7) (5,8) (6,11) (7,9) (8,11) (8,10) (8,9) (9,11) (9,10) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,8,9,10,11,6,3,4,2,7,1", "difficulty": "hard", "doc_id": "175"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,11) (0,12) (0,6) (0,2) (0,7) (0,1) (0,10) (0,9) (1,6) (1,9) (1,13) (1,8) (1,11) (1,7) (2,7) (2,8) (2,3) (2,4) (2,10) (2,5) (2,6) (2,13) (3,8) (3,4) (3,9) (3,7) (3,12) (3,10) (4,11) (4,12) (4,8) (4,6) (4,9) (5,10) (5,8) (5,9) (5,11) (5,12) (5,6) (6,10) (6,13) (6,12) (6,9) (7,10) (7,11) (7,8) (7,12) (7,9) (7,13) (8,9) (8,13) (8,11) (8,12) (8,10) (9,11) (9,12) (10,13) (10,12) (11,12) (11,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,12,11,13,10,8,7,3,4,2,5,6,1", "difficulty": "hard", "doc_id": "176"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,3) (0,1) (0,2) (1,4) (1,5) (2,3) (2,4) (3,5) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,4,1,5,3", "difficulty": "easy", "doc_id": "177"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,5) (0,4) (0,1) (1,3) (1,2) (1,5) (1,4) (2,3) (2,4) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,5,4,2,3", "difficulty": "easy", "doc_id": "178"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,5) (0,2) (0,1) (0,11) (0,7) (0,4) (1,10) (1,7) (1,14) (1,2) (2,6) (2,9) (3,6) (3,14) (4,12) (5,7) (5,6) (6,8) (6,10) (6,12) (6,7) (7,10) (7,14) (7,12) (7,9) (7,13) (8,9) (8,13) (9,10) (10,12) (11,14) (11,12) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,12,11,14,13,8,9,2,1,10,7,5,6,3", "difficulty": "hard", "doc_id": "179"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,7) (0,12) (0,15) (0,5) (0,9) (0,14) (0,2) (0,3) (0,13) (0,11) (0,16) (1,10) (1,8) (1,6) (1,16) (1,7) (1,4) (1,11) (1,15) (1,12) (1,5) (1,2) (2,8) (2,3) (2,12) (2,9) (2,11) (2,7) (2,15) (2,13) (2,6) (2,16) (3,13) (3,8) (3,15) (3,7) (3,6) (4,5) (4,15) (4,8) (4,13) (5,14) (5,15) (5,7) (5,16) (5,13) (5,6) (5,11) (6,13) (6,16) (6,10) (7,13) (7,8) (7,15) (8,15) (8,16) (8,11) (8,10) (8,14) (8,9) (9,11) (9,13) (9,12) (9,15) (9,10) (9,14) (10,13) (10,16) (11,15) (11,13) (12,13) (12,14) (12,16) (12,15) (13,15) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,16,14,12,15,13,11,9,10,8,7,5,6,3,2,1,4", "difficulty": "hard", "doc_id": "180"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,8) (0,9) (0,10) (0,12) (0,15) (0,5) (1,10) (1,15) (1,12) (1,9) (1,6) (1,14) (1,4) (1,2) (1,13) (1,5) (2,12) (2,9) (2,15) (2,10) (2,4) (2,13) (2,11) (2,7) (2,3) (2,14) (2,8) (3,12) (3,13) (3,10) (3,8) (3,11) (3,4) (3,15) (4,9) (4,7) (4,11) (4,5) (4,12) (4,10) (4,13) (4,8) (5,14) (5,11) (5,12) (5,8) (5,13) (6,14) (6,12) (6,11) (6,8) (6,7) (7,9) (7,10) (7,14) (7,15) (8,14) (8,15) (8,10) (8,12) (8,13) (9,13) (9,15) (9,14) (9,10) (10,14) (10,15) (11,14) (12,15) (12,14) (13,15) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,13,14,12,15,10,9,7,6,8,4,11,3,2,1", "difficulty": "hard", "doc_id": "181"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,7) (0,9) (0,10) (0,3) (0,1) (0,8) (0,4) (1,3) (1,12) (1,10) (2,8) (2,10) (2,5) (2,11) (2,12) (2,6) (2,7) (3,12) (3,9) (3,5) (3,7) (3,8) (3,4) (4,5) (4,6) (5,11) (5,12) (5,10) (5,8) (6,7) (6,9) (6,10) (6,11) (7,8) (7,10) (7,9) (8,9) (9,12) (9,10) (10,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,6,11,5,8,9,10,12,2,7,3,1", "difficulty": "hard", "doc_id": "182"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,9) (0,1) (0,4) (0,2) (1,11) (1,10) (1,2) (1,3) (1,5) (1,8) (2,8) (2,5) (2,9) (2,6) (2,3) (2,7) (2,11) (3,5) (3,10) (3,11) (3,7) (3,9) (3,4) (3,8) (4,7) (4,10) (4,11) (5,7) (5,9) (5,11) (5,10) (6,9) (6,11) (7,10) (7,9) (7,8) (8,11) (8,9) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,11,10,7,4,3,8,1,5,9,6", "difficulty": "hard", "doc_id": "183"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,6) (0,3) (0,2) (1,3) (1,4) (1,6) (2,3) (2,6) (2,5) (3,6) (3,4)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,4,1,6,2,5", "difficulty": "easy", "doc_id": "184"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,11) (0,4) (0,1) (0,5) (1,12) (1,5) (1,10) (2,7) (2,5) (2,4) (2,6) (3,6) (3,8) (4,9) (5,10) (6,12) (6,8) (7,8) (7,9) (7,11) (7,12) (8,12) (8,9) (9,11) (9,10) (9,12) (10,12) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,10,11,9,4,2,6,3,8,7,12,1", "difficulty": "hard", "doc_id": "185"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,8) (0,10) (0,4) (0,9) (0,11) (1,3) (1,5) (1,11) (1,10) (1,7) (1,9) (1,8) (2,8) (2,3) (2,9) (2,4) (2,10) (2,6) (2,5) (3,4) (3,11) (3,10) (3,5) (4,11) (4,6) (4,8) (4,5) (4,7) (4,9) (5,8) (5,6) (6,11) (6,8) (6,7) (6,10) (6,9) (7,8) (7,10) (7,11) (8,10) (8,11) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,8,10,9,6,7,4,5,2,3,1", "difficulty": "hard", "doc_id": "186"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,6) (0,9) (0,7) (0,10) (0,4) (0,3) (0,13) (0,1) (1,4) (1,6) (1,9) (1,12) (1,10) (1,7) (1,13) (2,3) (2,13) (2,11) (2,5) (2,9) (3,7) (3,9) (3,10) (3,8) (3,4) (3,12) (4,7) (4,6) (4,8) (4,5) (4,10) (5,13) (5,6) (5,7) (5,12) (6,7) (6,9) (6,8) (6,12) (7,8) (7,10) (7,13) (9,11) (9,10) (10,12) (10,13) (11,12) (11,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,13,11,12,10,9,6,8,7,5,4,3,2", "difficulty": "hard", "doc_id": "187"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,10) (0,12) (0,11) (0,7) (0,5) (0,16) (0,6) (0,3) (1,10) (1,13) (1,5) (1,16) (2,11) (2,12) (2,3) (2,7) (2,14) (2,10) (3,13) (3,9) (3,4) (3,6) (4,6) (4,5) (4,14) (4,16) (4,10) (5,15) (5,9) (5,16) (5,6) (5,13) (5,14) (5,12) (5,11) (5,8) (6,13) (6,7) (6,12) (6,9) (7,8) (7,15) (7,10) (7,11) (8,16) (8,14) (8,12) (9,14) (9,13) (10,13) (11,16) (11,12) (12,14) (12,15) (13,15) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,6,9,13,15,12,14,16,11,2,7,8,5,4,10,1", "difficulty": "hard", "doc_id": "188"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,6) (0,3) (1,7) (1,6) (1,3) (1,2) (2,4) (3,8) (3,6) (4,6) (5,6) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,8,7,1,2,4,6,5", "difficulty": "easy", "doc_id": "189"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,2) (1,4) (1,6) (1,2) (2,6) (2,5) (2,7) (3,6) (3,4) (3,7) (4,7) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,7,3,6,1,4,5", "difficulty": "easy", "doc_id": "190"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,8) (0,4) (0,5) (1,9) (1,11) (1,10) (1,6) (2,5) (2,10) (2,4) (2,9) (2,8) (2,11) (3,8) (3,11) (3,7) (3,9) (4,5) (4,9) (4,11) (5,6) (5,11) (6,8) (7,10) (7,8) (9,10) (9,11) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,11,10,7,3,9,4,2,8,6,1", "difficulty": "hard", "doc_id": "191"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,8) (0,4) (1,5) (1,2) (1,4) (2,6) (2,4) (2,8) (3,6) (3,4) (3,7) (3,5) (4,7) (4,8) (4,5) (5,6) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,5,6,3,7,8,2,1", "difficulty": "easy", "doc_id": "192"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,6) (0,2) (0,7) (0,4) (0,8) (1,5) (1,4) (1,8) (1,3) (2,6) (2,8) (3,7) (4,8) (4,5) (4,6) (4,7) (5,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,2,6,4,7,5,1,3", "difficulty": "easy", "doc_id": "193"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,6) (0,12) (0,9) (0,3) (0,8) (0,16) (0,5) (0,2) (0,18) (0,17) (0,1) (0,10) (0,7) (1,10) (1,4) (1,12) (1,3) (1,7) (1,16) (1,15) (1,17) (1,8) (2,11) (2,7) (2,8) (2,14) (2,3) (2,17) (2,5) (2,18) (2,13) (2,10) (2,4) (3,18) (3,8) (3,7) (3,13) (3,11) (3,9) (3,14) (3,12) (3,17) (4,10) (4,11) (4,15) (4,5) (4,16) (5,18) (5,8) (5,16) (5,12) (5,13) (5,6) (5,7) (5,14) (6,16) (6,13) (6,11) (6,10) (6,9) (7,9) (7,17) (7,12) (7,14) (7,10) (7,8) (7,13) (7,18) (7,11) (7,15) (8,17) (8,14) (8,11) (8,9) (8,16) (8,12) (8,13) (9,16) (9,18) (9,17) (9,14) (9,10) (9,13) (10,18) (10,11) (10,16) (10,17) (11,18) (11,16) (11,15) (11,12) (12,17) (12,13) (12,18) (12,15) (12,14) (13,18) (13,16) (13,17) (14,16) (14,17) (14,15) (15,16) (15,18) (15,17) (16,18) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,15,17,16,18,13,12,14,9,10,11,8,3,1,4,2,5,6", "difficulty": "hard", "doc_id": "194"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,9) (0,10) (0,3) (0,2) (0,1) (0,14) (0,5) (0,12) (0,4) (1,17) (1,12) (1,11) (1,19) (1,8) (1,2) (1,14) (2,6) (2,11) (2,15) (2,8) (2,18) (2,4) (2,17) (2,10) (2,5) (2,12) (2,13) (3,10) (3,16) (3,15) (3,14) (3,6) (3,18) (3,19) (3,9) (4,9) (4,7) (4,12) (5,17) (5,13) (5,19) (5,7) (5,6) (5,9) (5,8) (6,8) (6,10) (6,16) (6,19) (7,13) (7,14) (8,13) (8,16) (8,18) (8,14) (8,11) (9,11) (9,18) (9,10) (9,14) (10,18) (10,17) (10,11) (10,16) (10,13) (11,18) (11,16) (11,14) (12,13) (12,16) (13,19) (13,17) (14,17) (17,18) (18,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,12,16,11,14,17,18,19,13,10,9,3,15,2,1,8,6,5,7", "difficulty": "hard", "doc_id": "195"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,4) (0,2) (0,1) (1,3) (2,5) (2,4) (3,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,3,5,2,4", "difficulty": "easy", "doc_id": "196"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,9) (0,12) (0,4) (0,8) (0,5) (0,2) (0,10) (0,7) (0,11) (0,3) (0,13) (1,10) (1,13) (1,8) (1,3) (2,13) (2,7) (2,11) (2,4) (3,10) (3,4) (3,8) (3,11) (3,7) (3,9) (4,5) (4,12) (4,8) (5,11) (5,7) (5,9) (5,13) (5,6) (5,12) (6,11) (6,9) (6,8) (7,9) (7,8) (8,10) (8,9) (9,10) (10,13) (10,11) (11,12) (11,13) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,13,12,11,10,9,8,6,5,7,2,4,3,1", "difficulty": "hard", "doc_id": "197"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,9) (0,10) (0,2) (0,16) (0,4) (0,7) (0,11) (0,18) (0,5) (0,3) (0,12) (0,15) (1,18) (1,10) (1,3) (1,12) (1,11) (1,9) (1,2) (1,16) (1,6) (1,5) (2,10) (2,11) (2,5) (2,12) (2,14) (2,18) (2,4) (2,6) (2,9) (2,8) (3,12) (3,8) (3,16) (3,14) (3,6) (3,7) (3,5) (3,18) (3,4) (3,10) (3,15) (3,17) (4,6) (4,5) (4,17) (4,9) (4,11) (4,8) (4,16) (4,13) (4,15) (4,7) (4,14) (5,10) (5,12) (5,14) (5,9) (5,11) (5,16) (5,7) (5,17) (5,18) (5,8) (6,8) (6,18) (6,7) (6,10) (6,12) (6,13) (6,9) (7,15) (7,18) (7,17) (7,10) (7,9) (7,13) (8,16) (8,9) (8,10) (8,14) (8,15) (9,17) (9,13) (9,15) (9,12) (9,18) (10,12) (10,16) (10,15) (10,14) (11,14) (11,16) (11,18) (11,12) (11,15) (12,14) (12,17) (12,18) (12,15) (12,13) (13,17) (13,15) (13,16) (13,18) (14,16) (14,17) (15,18) (15,17) (15,16) (16,18) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,15,16,18,17,14,12,13,9,8,10,7,6,4,11,5,3,1,2", "difficulty": "hard", "doc_id": "198"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,13) (0,1) (0,12) (1,3) (1,2) (1,13) (1,8) (1,5) (2,6) (2,10) (2,8) (2,12) (3,13) (3,5) (3,11) (3,9) (3,6) (4,7) (4,11) (4,6) (4,9) (4,10) (5,12) (5,7) (6,12) (6,10) (7,8) (7,13) (8,10) (9,13) (10,12) (10,11) (11,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,10,11,13,9,4,6,3,5,7,8,2,1", "difficulty": "hard", "doc_id": "199"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,8) (0,2) (0,3) (0,6) (1,5) (1,4) (1,8) (2,4) (3,7) (3,4) (3,6) (4,8) (4,7) (5,6) (6,8) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,5,1,8,7,3,4,2", "difficulty": "easy", "doc_id": "200"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,4) (0,2) (1,3) (1,7) (1,6) (2,6) (2,5) (3,4) (3,5) (4,6) (4,5) (5,6) (5,8) (5,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,6,4,3,1,7,5,8", "difficulty": "easy", "doc_id": "201"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,9) (0,15) (0,11) (0,6) (0,4) (0,5) (0,17) (0,3) (0,10) (0,2) (1,12) (1,3) (1,15) (1,2) (1,10) (1,5) (1,14) (1,4) (1,11) (1,17) (1,6) (1,13) (2,9) (2,14) (2,5) (2,17) (2,7) (2,12) (2,6) (2,3) (3,10) (3,9) (3,8) (3,6) (3,17) (3,4) (3,12) (3,14) (3,13) (4,10) (4,8) (4,5) (4,9) (4,7) (4,6) (4,14) (4,17) (4,13) (5,15) (5,13) (5,10) (5,17) (5,9) (5,8) (6,17) (6,8) (6,7) (6,16) (6,15) (6,10) (6,13) (7,8) (7,9) (7,10) (7,12) (7,14) (8,16) (8,15) (8,14) (8,11) (8,9) (8,17) (8,12) (9,11) (9,10) (9,12) (10,17) (10,14) (10,11) (10,15) (10,13) (11,14) (11,17) (11,12) (12,15) (12,13) (12,14) (12,17) (13,15) (13,16) (14,15) (14,17) (14,16) (15,16) (15,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,3,13,16,15,17,14,12,11,10,9,8,7,6,4,5,1", "difficulty": "hard", "doc_id": "202"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,11) (0,6) (0,5) (0,4) (1,3) (1,9) (1,2) (1,10) (1,7) (2,11) (3,10) (3,6) (3,11) (4,9) (4,11) (5,10) (5,6) (5,11) (5,9) (6,9) (6,10) (7,8) (7,9) (8,9) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,11,10,3,6,5,9,8,7,1,2", "difficulty": "hard", "doc_id": "203"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,14) (0,5) (0,13) (0,8) (0,15) (0,4) (0,17) (0,10) (1,8) (1,10) (1,4) (1,11) (1,16) (2,17) (2,12) (2,7) (3,8) (3,13) (3,6) (3,15) (3,9) (4,8) (4,17) (4,16) (4,12) (4,10) (4,13) (5,9) (5,7) (5,12) (5,18) (5,13) (5,8) (6,15) (6,17) (6,12) (6,18) (6,11) (6,13) (6,9) (6,14) (6,7) (7,9) (7,11) (7,16) (7,17) (7,12) (7,15) (8,11) (8,16) (8,18) (9,11) (9,17) (10,17) (10,11) (10,14) (10,15) (10,13) (10,18) (11,16) (11,14) (11,13) (11,17) (11,12) (12,14) (12,15) (13,16) (13,15) (13,14) (14,15) (15,16) (15,18) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,18,17,11,12,15,16,8,1,4,13,14,6,3,9,5,7,2", "difficulty": "hard", "doc_id": "204"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,2) (0,3) (0,1) (0,11) (0,4) (0,9) (0,8) (1,5) (1,8) (1,9) (1,6) (1,3) (1,2) (2,9) (2,4) (2,11) (2,6) (2,7) (2,8) (3,8) (3,9) (3,11) (3,10) (3,7) (3,4) (3,5) (4,11) (4,8) (4,10) (4,9) (4,7) (5,11) (5,8) (5,9) (5,7) (6,11) (6,8) (6,10) (7,9) (7,8) (7,11) (8,9) (8,11) (9,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,11,9,7,5,3,4,10,6,2,1", "difficulty": "hard", "doc_id": "205"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,6) (0,4) (1,2) (1,3) (1,4) (2,4) (4,5) (4,6) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,5,4,2,1,3", "difficulty": "easy", "doc_id": "206"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,1) (0,2) (0,4) (0,5) (1,2) (1,4) (1,3) (2,3) (2,5) (3,5) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,4,1,3,2", "difficulty": "easy", "doc_id": "207"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,4) (0,2) (0,6) (0,1) (1,5) (1,3) (1,4) (2,4) (2,3) (2,5) (3,6) (3,5) (3,7) (3,4) (5,7) (6,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,4,3,6,7,5,2", "difficulty": "easy", "doc_id": "208"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,3) (0,2) (0,4) (1,4) (1,6) (1,3) (1,5) (2,6) (2,4) (2,3) (3,6) (3,5) (4,6) (4,5) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,5,6,2,3,1", "difficulty": "easy", "doc_id": "209"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,8) (0,4) (0,7) (0,2) (0,6) (1,4) (1,5) (1,6) (1,8) (1,2) (2,6) (2,4) (2,3) (2,9) (3,9) (3,8) (3,4) (4,7) (4,9) (4,6) (4,5) (5,7) (5,9) (6,7) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,7,5,9,8,3,4,2,1", "difficulty": "easy", "doc_id": "210"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,9) (0,2) (0,8) (0,4) (0,10) (0,5) (0,12) (1,12) (1,11) (1,4) (1,9) (1,3) (1,8) (1,2) (2,3) (2,4) (2,12) (2,7) (3,12) (3,6) (3,5) (4,5) (4,6) (4,7) (5,9) (5,10) (5,11) (5,6) (6,8) (6,11) (6,10) (7,8) (8,12) (8,11) (9,11) (10,12) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,10,11,9,5,6,8,7,4,2,3,1", "difficulty": "hard", "doc_id": "211"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,1) (0,3) (0,2) (0,7) (0,6) (1,4) (1,2) (1,7) (1,5) (1,6) (2,3) (2,4) (2,7) (2,5) (3,6) (3,4) (4,5) (4,6) (4,8) (5,7) (5,6) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,5,7,8,4,3,2,1", "difficulty": "easy", "doc_id": "212"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,7) (0,5) (0,6) (1,5) (1,7) (2,3) (2,6) (2,7) (3,4) (3,8) (4,5) (4,6) (4,8) (5,8) (5,6) (6,7) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,8,7,2,3,4,5,1", "difficulty": "easy", "doc_id": "213"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,7) (0,4) (0,1) (0,5) (0,3) (0,2) (1,7) (1,3) (1,6) (2,4) (2,6) (3,5) (3,6) (4,6) (4,7) (4,8) (5,8) (5,6) (5,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,6,5,8,4,7,1,3", "difficulty": "easy", "doc_id": "214"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,10) (0,2) (0,9) (1,3) (1,10) (2,9) (2,3) (2,8) (2,10) (2,7) (2,5) (2,6) (3,5) (3,10) (4,9) (4,5) (5,8) (5,10) (5,9) (6,10) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,4,5,8,7,2,6,10,3,1", "difficulty": "hard", "doc_id": "215"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,6) (0,5) (0,2) (0,9) (0,4) (0,7) (0,10) (0,3) (0,1) (1,2) (1,6) (1,3) (1,4) (1,8) (2,10) (2,4) (2,5) (2,6) (2,8) (2,7) (3,10) (3,5) (3,7) (3,8) (3,6) (4,5) (4,10) (4,8) (4,9) (5,9) (5,8) (5,6) (6,9) (6,7) (7,8) (8,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,8,10,4,9,6,7,3,5,2", "difficulty": "hard", "doc_id": "216"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,3) (0,1) (1,2) (1,7) (2,4) (2,3) (2,6) (3,8) (3,7) (3,6) (4,8) (4,5) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,7,3,8,4,5,6,2", "difficulty": "easy", "doc_id": "217"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,3) (0,12) (0,7) (0,10) (0,4) (0,11) (0,8) (1,2) (1,7) (1,8) (1,4) (1,5) (1,6) (1,10) (1,9) (2,7) (2,3) (2,6) (2,4) (2,10) (2,11) (2,9) (2,5) (3,9) (3,11) (3,10) (3,6) (3,7) (3,5) (4,12) (4,8) (4,10) (4,6) (4,7) (5,7) (5,10) (5,12) (5,6) (5,9) (6,11) (6,10) (6,12) (6,8) (7,10) (7,11) (7,9) (7,12) (8,11) (8,10) (8,9) (9,11) (9,10) (10,11) (10,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,9,10,12,7,11,6,5,3,2,4,1", "difficulty": "hard", "doc_id": "218"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,3) (0,18) (0,8) (0,6) (0,12) (1,6) (1,3) (1,18) (1,7) (1,16) (1,17) (1,13) (1,2) (1,15) (2,16) (2,6) (2,12) (2,11) (2,7) (2,14) (3,13) (3,10) (3,16) (3,14) (3,17) (3,12) (4,10) (4,8) (4,9) (4,5) (4,15) (4,17) (5,12) (5,10) (5,7) (5,17) (5,18) (6,17) (6,14) (6,12) (6,18) (7,11) (7,15) (7,17) (8,13) (8,14) (9,10) (9,12) (9,15) (9,18) (9,16) (10,12) (10,16) (10,11) (10,18) (12,16) (13,15) (13,16) (14,16) (14,18) (14,17) (14,15) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,16,17,14,15,13,8,4,5,18,9,10,11,7,2,6,1,3", "difficulty": "hard", "doc_id": "219"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,10) (0,3) (0,13) (0,12) (0,11) (0,6) (0,8) (1,10) (1,8) (1,14) (1,4) (1,13) (1,6) (1,5) (1,15) (1,11) (1,7) (2,13) (2,10) (2,4) (2,5) (2,8) (2,11) (2,3) (2,6) (2,15) (2,7) (3,8) (3,14) (3,4) (3,9) (3,5) (3,11) (3,12) (4,10) (4,15) (4,9) (4,6) (4,8) (4,7) (4,5) (5,11) (5,15) (5,10) (5,6) (6,10) (6,11) (6,8) (6,13) (6,12) (7,14) (7,8) (7,12) (7,13) (7,15) (7,10) (7,11) (8,13) (8,11) (8,14) (9,15) (9,10) (9,14) (10,15) (10,14) (10,13) (11,13) (11,12) (12,13) (12,15) (13,15) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,14,15,13,12,11,7,10,9,4,5,3,2,6,1", "difficulty": "hard", "doc_id": "220"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,2) (0,1) (1,6) (1,9) (1,2) (1,4) (1,10) (2,9) (3,10) (3,6) (3,9) (3,5) (4,8) (4,11) (4,6) (5,8) (6,7) (7,9) (7,10) (8,9) (9,10) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,10,11,4,8,5,3,6,7,9,2", "difficulty": "hard", "doc_id": "221"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,3) (0,5) (0,1) (1,8) (1,5) (1,7) (1,6) (1,2) (1,4) (2,5) (2,8) (2,6) (2,7) (2,3) (3,6) (3,8) (3,5) (4,6) (4,5) (4,8) (5,7) (5,6) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,4,8,7,6,5,3,2", "difficulty": "easy", "doc_id": "222"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,5) (0,1) (1,5) (1,8) (2,5) (2,8) (3,6) (3,8) (4,7) (4,6) (5,7) (5,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,8,3,6,4,7,5,2", "difficulty": "easy", "doc_id": "223"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,3) (0,1) (1,2) (2,3) (3,4)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,2,3,4", "difficulty": "easy", "doc_id": "224"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,16) (0,8) (0,4) (0,6) (0,9) (0,1) (0,14) (1,3) (1,8) (1,14) (1,6) (1,16) (1,13) (1,7) (2,3) (2,14) (2,15) (2,16) (2,10) (2,7) (2,5) (3,15) (3,16) (3,8) (3,17) (3,7) (3,5) (4,17) (4,10) (4,13) (5,10) (5,11) (5,9) (6,10) (6,16) (6,17) (7,16) (7,17) (7,10) (7,14) (8,12) (8,16) (8,14) (8,11) (9,10) (9,12) (9,11) (9,17) (10,14) (10,13) (10,12) (11,15) (11,12) (12,15) (12,13) (12,16) (13,15) (14,17) (14,15) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,14,16,12,13,15,11,9,17,7,2,5,3,8,1,6,10,4", "difficulty": "hard", "doc_id": "225"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,5) (0,15) (0,1) (0,6) (0,9) (0,10) (1,14) (1,5) (1,11) (1,6) (1,16) (1,8) (1,7) (2,8) (2,9) (2,11) (2,6) (3,9) (3,5) (3,6) (3,16) (3,15) (3,8) (3,13) (4,8) (4,12) (4,13) (4,14) (4,15) (4,10) (5,15) (5,13) (5,11) (6,15) (6,7) (6,11) (6,10) (7,12) (7,14) (7,10) (7,9) (7,16) (7,8) (8,10) (8,12) (8,9) (9,10) (9,13) (9,14) (9,15) (9,16) (10,16) (10,11) (10,13) (11,15) (11,13) (12,16) (13,14) (13,15) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,13,15,11,6,7,8,12,4,14,16,1,5,3,9,2", "difficulty": "hard", "doc_id": "226"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,4) (0,9) (0,2) (0,1) (1,5) (1,10) (1,3) (1,7) (1,6) (2,7) (2,6) (3,6) (3,5) (4,6) (4,5) (4,7) (4,8) (4,10) (5,8) (5,6) (6,7) (7,9) (7,8) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,6,3,5,8,4,10,9,7,2", "difficulty": "hard", "doc_id": "227"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,1) (0,10) (0,2) (0,7) (0,8) (0,11) (0,3) (1,2) (1,6) (2,7) (2,8) (2,6) (3,9) (3,7) (3,10) (3,5) (3,4) (4,7) (4,8) (4,6) (4,10) (5,6) (5,7) (5,9) (5,8) (6,8) (6,11) (7,8) (8,10) (8,9) (9,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,6,1,2,8,10,4,7,5,3,9,12", "difficulty": "hard", "doc_id": "228"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,13) (0,15) (0,2) (1,7) (1,14) (1,9) (2,13) (2,7) (2,3) (2,8) (2,15) (2,14) (2,12) (3,10) (3,14) (3,5) (3,7) (3,8) (3,11) (4,5) (4,12) (4,10) (4,14) (4,11) (5,6) (5,15) (5,11) (5,7) (6,8) (6,14) (6,15) (7,11) (7,13) (7,15) (7,14) (7,10) (7,9) (8,12) (8,13) (8,10) (8,15) (8,11) (9,11) (9,12) (10,13) (11,12) (12,14) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,12,13,10,8,11,9,1,14,6,15,7,3,5,4", "difficulty": "hard", "doc_id": "229"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,2) (0,7) (0,1) (0,13) (0,9) (0,6) (1,10) (1,5) (1,8) (1,4) (1,9) (1,13) (1,12) (1,7) (1,3) (2,8) (2,12) (2,5) (2,3) (2,7) (2,11) (3,9) (3,10) (3,7) (3,5) (3,11) (4,9) (4,11) (4,8) (4,13) (4,15) (4,10) (5,15) (5,8) (5,11) (5,6) (6,13) (6,8) (7,9) (7,11) (7,10) (7,13) (8,10) (8,9) (9,12) (9,14) (10,14) (10,11) (11,15) (11,14) (11,12) (12,14) (12,13) (13,15) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,8,9,14,15,13,12,11,10,7,3,2,5,1,4", "difficulty": "hard", "doc_id": "230"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,1) (0,8) (0,5) (1,4) (1,3) (2,9) (2,7) (2,4) (2,3) (2,6) (3,6) (3,4) (3,7) (3,8) (4,5) (4,7) (5,6) (5,8) (5,9) (6,8) (6,9) (6,7) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,9,8,6,7,4,2,3,1", "difficulty": "easy", "doc_id": "231"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,6) (0,3) (0,8) (0,2) (0,1) (0,4) (0,7) (1,5) (1,7) (1,6) (1,8) (2,3) (2,8) (2,5) (2,7) (2,6) (3,8) (3,4) (3,7) (4,8) (5,8) (5,6) (6,7) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,8,5,1,6,2,3,4", "difficulty": "easy", "doc_id": "232"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (0,1) (0,4) (1,4) (1,7) (2,3) (2,8) (2,6) (2,9) (3,4) (3,7) (3,5) (3,6) (3,9) (3,8) (4,6) (4,9) (4,7) (5,9) (5,8) (5,7) (6,8) (7,9) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,9,8,6,2,3,5,7,1", "difficulty": "easy", "doc_id": "233"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,9) (0,2) (0,3) (0,7) (1,5) (1,6) (1,7) (1,2) (2,8) (2,9) (2,6) (2,4) (2,3) (3,9) (4,7) (4,6) (4,8) (4,5) (5,7) (5,6) (6,7) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,9,3,2,6,1,5,4,8", "difficulty": "easy", "doc_id": "234"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,15) (0,17) (0,7) (0,14) (0,8) (0,3) (0,4) (1,3) (1,10) (1,2) (1,16) (1,8) (1,15) (1,5) (1,4) (1,9) (1,7) (1,18) (1,17) (1,12) (2,18) (2,13) (2,8) (2,15) (2,3) (2,11) (2,16) (2,17) (3,7) (3,5) (3,12) (3,18) (3,8) (3,9) (3,10) (3,15) (3,16) (3,11) (3,6) (4,9) (4,18) (4,10) (4,17) (4,16) (4,15) (4,14) (4,11) (4,6) (4,5) (4,8) (4,12) (4,13) (5,17) (5,15) (5,14) (5,16) (5,18) (5,10) (5,9) (5,6) (5,8) (5,7) (6,16) (6,14) (6,18) (6,10) (6,8) (6,15) (6,12) (6,7) (7,10) (7,18) (7,9) (7,8) (8,18) (8,17) (8,10) (8,15) (8,12) (8,11) (9,18) (9,15) (9,12) (9,16) (9,13) (9,11) (9,17) (10,17) (10,11) (10,18) (10,15) (11,14) (11,16) (11,13) (12,13) (12,14) (12,15) (13,14) (13,15) (13,18) (13,16) (14,17) (14,18) (14,16) (14,15) (15,16) (15,18) (16,18) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,13,16,18,17,14,15,12,9,11,10,8,7,6,5,3,2,1", "difficulty": "hard", "doc_id": "235"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,1) (0,7) (0,5) (1,4) (2,8) (2,4) (2,5) (2,6) (2,7) (3,7) (3,8) (4,5) (4,6) (4,9) (4,7) (5,8) (5,7) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,4,9,7,3,8,5,2,6", "difficulty": "easy", "doc_id": "236"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,6) (0,5) (1,9) (1,6) (1,2) (1,7) (2,7) (2,9) (2,8) (3,6) (3,7) (3,4) (4,7) (4,6) (5,9) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,9,7,4,3,6,1,2,8", "difficulty": "easy", "doc_id": "237"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,8) (0,12) (0,4) (0,6) (0,9) (0,10) (0,11) (1,11) (1,4) (1,10) (2,4) (2,7) (2,13) (2,9) (2,8) (2,5) (2,6) (3,5) (3,4) (3,13) (3,9) (3,6) (3,12) (3,10) (4,10) (4,11) (4,7) (4,9) (4,8) (4,6) (4,12) (5,9) (5,6) (5,8) (5,10) (5,11) (6,13) (6,8) (6,10) (6,7) (6,12) (7,13) (7,8) (8,10) (9,11) (9,10) (10,13) (11,12)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,12,6,7,8,10,13,3,9,5,2,4,1", "difficulty": "hard", "doc_id": "238"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,12) (0,8) (0,5) (0,14) (0,2) (0,9) (0,10) (0,15) (0,7) (0,16) (1,15) (1,2) (1,11) (1,3) (1,10) (2,11) (3,15) (3,16) (3,10) (3,7) (3,13) (3,17) (4,12) (4,5) (4,17) (4,10) (4,14) (4,11) (5,15) (5,8) (5,6) (6,9) (6,15) (6,14) (7,12) (7,17) (7,10) (7,16) (8,10) (8,13) (8,17) (8,9) (9,10) (9,17) (9,16) (10,11) (10,13) (10,16) (10,17) (11,12) (11,16) (12,15) (12,17) (13,15) (13,17) (14,16) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,16,17,13,15,12,11,4,14,6,5,8,9,10,7,3,1,2", "difficulty": "hard", "doc_id": "239"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,16) (0,12) (0,4) (0,13) (0,6) (0,17) (0,8) (0,2) (0,15) (1,13) (1,15) (1,8) (1,3) (1,10) (1,17) (2,16) (2,15) (2,6) (2,11) (2,9) (2,12) (2,8) (2,14) (2,17) (2,13) (2,18) (2,3) (3,12) (3,4) (3,13) (3,16) (3,15) (3,17) (3,11) (3,10) (3,9) (4,12) (4,7) (4,18) (4,5) (4,14) (4,8) (4,13) (4,10) (4,9) (5,13) (5,17) (5,14) (5,12) (5,7) (5,10) (5,18) (6,7) (6,17) (6,16) (6,15) (6,14) (6,13) (6,9) (7,9) (7,13) (7,16) (7,12) (7,11) (7,10) (8,13) (8,11) (8,16) (8,18) (8,10) (8,17) (8,9) (9,18) (9,17) (9,10) (9,13) (9,12) (9,16) (9,14) (10,13) (10,18) (10,16) (10,17) (10,11) (11,18) (11,16) (11,17) (11,12) (12,14) (12,15) (12,17) (13,15) (13,18) (13,17) (14,15) (14,16) (15,18) (15,16) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,15,16,17,13,18,11,12,14,9,10,8,4,5,7,6,2,3,1", "difficulty": "hard", "doc_id": "240"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,6) (0,1) (0,2) (0,13) (0,3) (0,9) (0,5) (0,18) (1,9) (1,11) (1,14) (1,10) (1,8) (1,13) (1,12) (2,12) (2,11) (2,17) (2,16) (2,13) (3,11) (3,8) (3,12) (3,6) (3,13) (3,15) (3,14) (3,10) (4,12) (4,18) (4,7) (4,9) (5,12) (5,10) (5,16) (5,17) (5,13) (6,16) (6,7) (6,11) (7,9) (7,17) (7,8) (7,15) (8,17) (8,9) (8,13) (9,11) (10,17) (10,15) (11,15) (11,12) (11,18) (12,17) (12,15) (13,16) (13,15) (13,14) (14,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,18,11,12,15,13,14,17,8,3,6,7,4,9,1,10,5,16,2", "difficulty": "hard", "doc_id": "241"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,2) (0,8) (0,4) (0,3) (1,4) (1,7) (1,5) (2,6) (2,5) (2,4) (2,8) (3,6) (3,5) (4,5) (5,6) (5,7) (5,8) (6,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,5,8,6,2,4,1,7", "difficulty": "easy", "doc_id": "242"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,5) (0,6) (0,1) (0,4) (1,4) (1,3) (1,5) (2,4) (2,3) (4,6) (4,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,4,5,1,3,2", "difficulty": "easy", "doc_id": "243"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,10) (0,11) (0,1) (1,13) (1,5) (2,9) (2,12) (2,6) (2,4) (2,10) (2,7) (3,9) (3,13) (3,7) (3,10) (4,10) (4,6) (5,6) (6,11) (7,8) (7,11) (7,9) (7,10) (7,13) (8,9) (11,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,5,6,11,13,7,8,9,3,10,4,2,12", "difficulty": "hard", "doc_id": "244"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,12) (0,7) (0,11) (0,2) (0,10) (0,6) (0,3) (1,10) (1,11) (1,12) (1,2) (1,4) (1,8) (1,3) (1,9) (1,7) (2,12) (2,6) (2,8) (3,4) (3,7) (3,9) (3,8) (3,13) (3,6) (3,5) (3,10) (4,9) (4,12) (4,5) (4,6) (4,11) (4,13) (5,8) (5,12) (6,9) (6,12) (6,8) (6,7) (7,12) (7,9) (7,11) (7,8) (7,10) (8,13) (8,11) (9,12) (10,12) (10,11) (11,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,10,11,13,8,7,9,12,6,2,1,4,5", "difficulty": "hard", "doc_id": "245"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,4) (0,12) (0,1) (0,8) (0,2) (0,14) (0,9) (0,16) (0,5) (0,10) (0,6) (1,16) (1,15) (1,3) (1,17) (1,7) (1,8) (1,10) (1,6) (1,13) (1,12) (2,6) (2,17) (2,5) (2,4) (2,15) (2,8) (2,3) (3,13) (3,9) (3,16) (3,6) (3,15) (3,8) (3,14) (3,11) (3,10) (4,9) (4,17) (4,8) (4,10) (4,15) (4,5) (4,6) (5,15) (5,14) (5,13) (5,10) (5,6) (5,8) (5,11) (6,12) (6,14) (6,10) (6,13) (6,11) (6,16) (6,9) (7,9) (7,14) (7,11) (7,13) (7,16) (7,10) (7,17) (7,15) (8,14) (8,13) (8,15) (9,11) (9,16) (9,13) (9,12) (9,17) (10,13) (10,12) (10,17) (10,16) (11,17) (11,13) (11,14) (12,17) (12,16) (12,14) (12,15) (13,14) (14,15) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,9,17,16,12,15,14,13,11,7,10,5,8,4,2,3,1", "difficulty": "hard", "doc_id": "246"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,3) (0,16) (0,8) (0,5) (0,14) (0,13) (0,2) (0,10) (0,6) (0,11) (0,9) (1,8) (1,5) (1,3) (1,12) (1,14) (1,16) (1,2) (1,7) (2,8) (2,5) (2,14) (2,10) (2,6) (2,12) (2,11) (2,13) (2,3) (3,12) (3,6) (3,13) (3,16) (3,9) (3,7) (3,8) (3,5) (4,15) (4,16) (4,13) (4,7) (4,9) (4,8) (4,5) (5,12) (5,13) (5,14) (5,16) (6,12) (6,11) (6,15) (6,14) (6,9) (7,14) (7,9) (7,12) (7,8) (7,15) (8,11) (8,15) (8,16) (8,14) (9,10) (9,16) (10,11) (10,13) (10,16) (10,12) (10,14) (10,15) (11,13) (11,16) (12,14) (13,16) (14,15) (15,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,16,15,14,12,10,13,11,8,7,4,5,3,6,2,1", "difficulty": "hard", "doc_id": "247"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,12) (0,4) (0,10) (0,9) (0,8) (0,5) (1,2) (1,3) (1,12) (2,10) (2,9) (2,3) (2,12) (2,4) (2,5) (3,11) (3,8) (3,5) (3,10) (3,12) (3,4) (4,5) (4,7) (4,6) (4,11) (4,8) (5,8) (5,7) (5,9) (5,6) (6,11) (7,10) (7,11) (7,12) (8,11) (8,12) (9,12) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,6,11,10,7,12,9,2,4,8,3,1", "difficulty": "hard", "doc_id": "248"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,15) (0,18) (0,12) (0,5) (0,11) (0,4) (0,8) (0,1) (0,6) (1,18) (1,13) (1,6) (1,9) (1,11) (1,3) (1,8) (1,17) (1,7) (1,2) (1,12) (1,4) (2,17) (2,3) (2,11) (2,12) (2,7) (2,5) (2,9) (2,14) (2,4) (2,8) (3,4) (3,13) (3,10) (3,7) (3,8) (3,6) (3,9) (3,15) (3,14) (4,17) (4,6) (4,8) (4,9) (4,14) (4,16) (5,6) (5,7) (5,15) (5,17) (5,16) (5,10) (5,13) (6,13) (6,16) (6,14) (6,9) (7,18) (7,10) (7,16) (7,11) (7,8) (7,9) (8,11) (8,16) (8,17) (8,18) (9,11) (9,14) (9,13) (9,15) (9,12) (10,13) (10,16) (10,14) (10,11) (11,18) (11,16) (11,17) (11,13) (11,12) (12,13) (12,17) (12,16) (13,15) (13,16) (13,14) (14,17) (14,18) (14,16) (14,15) (15,18) (15,16) (15,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,9,12,16,15,17,14,18,11,13,10,7,8,4,3,1,2,5", "difficulty": "hard", "doc_id": "249"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,3) (0,9) (0,8) (0,4) (0,1) (0,2) (1,4) (1,7) (1,3) (1,2) (1,8) (2,4) (2,6) (3,8) (3,6) (3,9) (3,4) (3,5) (3,7) (4,9) (4,7) (4,6) (5,7) (5,8) (6,8) (6,9) (7,9) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,6,9,8,7,5,3,4,1", "difficulty": "easy", "doc_id": "250"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,11) (0,13) (0,14) (0,5) (0,8) (0,3) (0,9) (0,7) (1,3) (1,4) (1,2) (1,14) (1,13) (1,5) (2,12) (2,9) (2,4) (3,10) (3,14) (3,7) (3,11) (3,13) (4,8) (4,14) (4,7) (4,11) (4,5) (4,10) (5,11) (5,13) (5,10) (5,9) (5,8) (6,14) (6,7) (6,13) (6,11) (6,8) (7,10) (8,9) (8,13) (8,11) (8,14) (9,14) (9,12) (9,13) (9,11) (9,10) (10,14) (10,12) (10,11) (11,13) (11,12) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,10,11,12,9,13,14,6,8,5,4,2,1,3", "difficulty": "hard", "doc_id": "251"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 13, and the edges are: (0,3) (0,4) (0,1) (0,5) (0,13) (0,9) (0,2) (0,6) (0,10) (0,7) (1,6) (1,10) (1,8) (2,6) (2,4) (2,8) (2,9) (2,3) (3,7) (3,13) (3,6) (3,4) (3,10) (3,8) (3,5) (4,13) (4,7) (4,11) (4,9) (4,6) (4,5) (5,7) (5,6) (5,11) (5,10) (5,8) (6,12) (6,7) (6,13) (7,9) (7,11) (7,12) (8,10) (8,11) (8,13) (8,9) (9,13) (9,11) (9,12) (9,10) (10,13) (10,12) (11,13) (11,12) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,7,12,13,11,9,10,8,5,4,3,2,6,1", "difficulty": "hard", "doc_id": "252"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,4) (0,1) (0,2) (0,8) (0,9) (0,7) (0,16) (0,3) (0,12) (1,12) (1,5) (1,7) (1,2) (1,14) (1,6) (1,15) (1,11) (1,9) (2,12) (2,10) (3,14) (3,9) (3,11) (3,12) (3,15) (3,4) (3,16) (3,6) (4,11) (4,13) (4,5) (4,12) (4,7) (4,9) (4,8) (4,14) (4,6) (4,15) (5,8) (5,12) (5,11) (5,14) (6,13) (6,16) (6,12) (7,13) (8,12) (8,11) (8,16) (8,15) (8,9) (9,10) (9,14) (9,13) (9,12) (10,11) (10,12) (10,13) (10,14) (11,16) (11,14) (11,12) (11,15) (11,13) (12,13) (13,15) (14,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,12,13,15,11,14,16,8,5,4,6,3,9,10,2,1,7", "difficulty": "hard", "doc_id": "253"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,12) (0,2) (0,14) (0,1) (0,3) (0,10) (0,8) (0,7) (0,6) (0,11) (1,8) (1,12) (1,2) (1,9) (1,13) (1,15) (1,7) (2,8) (2,13) (2,6) (2,7) (2,12) (2,10) (2,11) (3,4) (3,6) (3,12) (3,7) (3,15) (3,13) (3,14) (4,11) (4,13) (4,10) (4,8) (4,5) (4,12) (4,6) (5,7) (5,14) (5,6) (5,12) (5,8) (5,13) (6,7) (6,10) (6,13) (6,14) (6,15) (7,13) (7,10) (7,14) (8,9) (8,10) (8,15) (8,11) (9,15) (9,14) (9,10) (9,12) (10,14) (10,12) (10,13) (11,15) (11,12) (11,13) (13,14) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,13,14,15,9,12,10,8,5,6,4,3,7,2,1", "difficulty": "hard", "doc_id": "254"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,6) (0,1) (0,3) (0,2) (0,5) (0,8) (0,7) (0,9) (1,2) (1,4) (1,5) (1,6) (1,9) (2,7) (2,5) (2,6) (2,8) (2,3) (2,9) (2,4) (3,9) (3,6) (3,7) (4,7) (4,6) (5,8) (5,7) (5,9) (6,8) (6,7) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,8,6,7,5,1,4,2,3", "difficulty": "easy", "doc_id": "255"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,6) (0,11) (0,10) (0,3) (0,12) (0,4) (0,7) (0,14) (1,5) (1,3) (1,4) (1,9) (1,13) (1,11) (2,9) (2,12) (2,6) (2,14) (2,8) (2,5) (2,7) (2,10) (2,4) (3,9) (3,4) (3,10) (3,13) (3,5) (3,7) (3,12) (4,6) (4,14) (4,12) (4,8) (4,7) (4,13) (4,9) (5,10) (5,12) (5,14) (5,13) (5,8) (6,7) (6,11) (6,10) (6,13) (6,8) (7,10) (7,13) (7,8) (8,10) (8,13) (8,9) (8,14) (9,11) (9,13) (9,10) (9,12) (10,11) (10,12) (10,14) (11,12) (11,14) (11,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,14,11,13,9,12,10,8,7,6,4,3,1,5,2", "difficulty": "hard", "doc_id": "256"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,8) (0,2) (0,3) (0,9) (0,6) (0,5) (0,1) (1,7) (1,2) (1,5) (1,9) (1,4) (2,3) (2,4) (2,8) (2,9) (3,8) (4,7) (4,5) (4,9) (4,8) (5,6) (5,8) (5,7) (6,9) (6,7) (6,8) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,4,8,7,5,6,9,2,3", "difficulty": "easy", "doc_id": "257"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,6) (0,4) (0,2) (1,6) (1,5) (2,6) (3,4) (3,5) (4,6) (4,5) (5,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,6,4,3,5,1", "difficulty": "easy", "doc_id": "258"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,6) (0,4) (0,14) (0,3) (0,19) (0,9) (0,16) (1,15) (1,7) (1,17) (1,14) (1,11) (1,8) (1,4) (1,5) (2,3) (2,6) (2,4) (2,12) (2,16) (2,19) (2,5) (3,4) (3,11) (3,12) (3,15) (3,13) (3,14) (3,5) (3,9) (3,16) (3,19) (3,8) (3,17) (4,12) (4,18) (4,5) (4,6) (4,7) (4,14) (5,6) (5,15) (5,12) (5,16) (5,18) (5,19) (6,19) (6,16) (6,17) (6,13) (6,7) (6,11) (7,12) (7,9) (8,13) (8,9) (8,19) (9,11) (9,15) (10,17) (10,14) (11,15) (11,12) (12,14) (12,17) (12,16) (13,19) (13,16) (13,17) (13,14) (14,19) (14,18) (14,17) (14,15) (15,17) (15,18) (15,16) (17,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,16,15,18,14,19,13,8,9,11,12,7,6,5,3,2,4,1,17,10", "difficulty": "hard", "doc_id": "259"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,2) (0,3) (0,17) (0,18) (0,11) (0,1) (0,14) (0,8) (0,5) (1,18) (1,10) (1,17) (1,15) (1,12) (1,8) (1,4) (1,2) (1,6) (1,7) (1,19) (1,3) (2,19) (2,7) (2,9) (2,14) (2,6) (2,12) (2,4) (2,10) (2,18) (2,5) (2,3) (2,15) (2,11) (2,17) (3,18) (3,8) (3,5) (3,11) (3,12) (3,6) (3,13) (4,7) (4,14) (4,19) (4,12) (4,9) (4,5) (4,6) (4,10) (4,16) (4,11) (4,18) (4,17) (5,15) (5,11) (5,13) (5,6) (5,7) (5,12) (5,17) (5,8) (5,14) (6,19) (6,10) (6,14) (6,12) (6,8) (6,18) (6,13) (7,17) (7,9) (7,10) (7,14) (7,13) (7,11) (7,19) (7,15) (7,18) (8,9) (8,17) (8,11) (8,18) (8,13) (8,12) (9,14) (9,18) (9,16) (9,15) (10,16) (10,18) (10,19) (10,15) (11,12) (11,19) (11,18) (11,14) (12,17) (12,18) (12,14) (12,15) (13,17) (13,19) (13,15) (13,18) (14,19) (14,18) (14,17) (15,16) (15,17) (15,18) (16,17) (16,19) (16,18) (17,19) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,14,17,18,16,19,13,15,12,11,8,9,7,10,6,4,2,3,1", "difficulty": "hard", "doc_id": "260"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,6) (0,10) (0,15) (0,5) (0,1) (0,17) (0,14) (0,12) (0,2) (0,11) (0,16) (0,4) (1,16) (1,13) (1,6) (1,10) (1,7) (1,11) (2,9) (2,13) (3,8) (3,14) (3,9) (3,4) (3,17) (3,7) (3,10) (3,11) (3,5) (4,16) (4,5) (4,11) (4,9) (4,6) (4,8) (4,12) (4,17) (4,13) (5,14) (5,7) (5,13) (5,16) (5,15) (5,6) (5,17) (6,8) (6,15) (6,16) (6,13) (6,11) (6,12) (7,9) (7,16) (7,17) (7,13) (7,15) (8,14) (8,13) (9,17) (9,15) (9,16) (9,14) (9,12) (10,17) (10,11) (10,16) (10,12) (11,14) (11,17) (11,13) (12,15) (12,16) (12,14) (13,15) (13,16) (13,17) (14,17) (14,15) (15,16) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,13,17,16,15,14,12,10,11,1,7,5,6,8,3,9,2", "difficulty": "hard", "doc_id": "261"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,6) (0,8) (0,1) (0,2) (0,3) (0,9) (1,8) (1,2) (1,7) (1,5) (1,9) (1,3) (2,6) (2,9) (2,4) (2,8) (3,9) (3,5) (3,6) (3,7) (4,7) (4,8) (4,9) (5,8) (5,9) (6,7) (6,8) (6,9) (7,9) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,8,7,6,3,5,1,2,4", "difficulty": "easy", "doc_id": "262"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,1) (0,7) (0,3) (0,6) (0,2) (0,4) (0,5) (1,5) (1,2) (1,4) (1,3) (1,6) (2,7) (2,6) (2,5) (2,4) (3,6) (3,4) (3,7) (4,6) (4,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,5,2,4,7,3,6,1", "difficulty": "easy", "doc_id": "263"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,9) (0,1) (0,16) (0,4) (0,6) (0,7) (0,10) (0,5) (0,14) (0,3) (0,12) (0,11) (0,13) (1,13) (1,9) (1,11) (1,8) (1,16) (1,14) (1,15) (1,3) (2,14) (2,8) (2,4) (2,13) (2,9) (2,6) (2,5) (2,16) (3,14) (3,13) (3,10) (3,15) (3,12) (3,8) (3,16) (3,5) (3,9) (4,8) (4,16) (4,9) (4,13) (4,12) (5,9) (5,11) (5,12) (5,16) (5,15) (5,14) (6,8) (6,7) (7,15) (7,8) (7,12) (8,14) (8,9) (8,10) (8,12) (8,11) (8,16) (9,10) (9,13) (9,15) (9,11) (10,16) (11,15) (11,14) (11,12) (12,14) (12,15) (13,14) (13,16) (13,15) (14,16) (14,15) (15,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,13,15,16,14,12,11,5,9,10,3,1,8,7,6,2,4", "difficulty": "hard", "doc_id": "264"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,7) (0,8) (1,9) (1,7) (2,5) (2,4) (3,7) (3,6) (4,7) (4,9) (5,6) (6,9) (7,9) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,9,4,2,5,6,3,7,1", "difficulty": "easy", "doc_id": "265"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,1) (0,3) (1,4) (1,3) (2,4) (3,4)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,1,4,2", "difficulty": "easy", "doc_id": "266"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,15) (0,16) (0,1) (0,5) (0,6) (0,12) (0,4) (0,14) (1,15) (1,11) (1,4) (1,3) (1,9) (1,12) (1,2) (1,5) (2,5) (2,3) (2,8) (2,13) (2,7) (3,7) (3,11) (3,14) (3,16) (3,9) (4,11) (4,16) (4,8) (4,9) (5,11) (6,11) (6,8) (7,9) (7,10) (7,12) (7,13) (7,14) (8,14) (8,12) (8,16) (8,10) (9,14) (9,13) (10,12) (10,14) (10,15) (11,12) (11,15) (12,13) (12,16) (14,16) (14,15) (15,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,14,15,16,12,13,9,7,10,8,6,11,5,2,3,1,4", "difficulty": "hard", "doc_id": "267"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,15) (0,14) (0,5) (0,7) (0,4) (0,1) (0,12) (0,10) (0,11) (0,3) (0,8) (1,7) (1,11) (1,8) (1,17) (1,15) (1,13) (1,9) (1,10) (1,2) (1,4) (2,8) (2,5) (2,14) (2,6) (2,13) (2,4) (2,12) (2,15) (2,7) (2,9) (2,10) (2,11) (2,17) (3,16) (3,13) (3,6) (3,17) (3,11) (3,10) (3,12) (3,15) (3,5) (3,8) (3,14) (4,7) (4,14) (4,10) (4,16) (4,12) (4,17) (4,13) (4,6) (4,15) (4,11) (4,5) (5,10) (5,9) (5,14) (5,6) (5,17) (5,16) (5,12) (5,7) (6,7) (6,13) (6,11) (6,10) (6,14) (6,8) (6,15) (7,8) (7,9) (7,12) (7,15) (7,14) (7,10) (8,15) (8,17) (8,9) (8,16) (8,10) (8,13) (8,11) (9,10) (9,11) (9,16) (9,15) (9,13) (10,13) (10,12) (10,14) (11,17) (11,13) (11,16) (11,15) (11,12) (12,15) (12,13) (12,16) (13,15) (13,17) (13,14) (14,16) (14,17) (15,16) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,11,12,16,17,14,13,15,9,10,7,6,4,1,2,5,3", "difficulty": "hard", "doc_id": "268"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,7) (0,5) (0,8) (0,6) (1,2) (1,5) (1,6) (1,4) (2,8) (2,4) (3,4) (3,10) (3,7) (4,8) (4,11) (4,10) (4,9) (4,6) (4,7) (5,8) (6,9) (6,11) (7,11) (7,10) (9,11) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,9,11,10,7,3,4,8,5,1,2", "difficulty": "hard", "doc_id": "269"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 16, and the edges are: (0,4) (0,16) (0,11) (0,12) (0,13) (0,10) (0,3) (0,1) (0,6) (1,8) (1,6) (1,10) (1,12) (1,14) (1,2) (1,13) (2,10) (2,14) (2,8) (2,3) (2,15) (2,12) (2,16) (2,4) (3,4) (3,16) (3,6) (3,10) (3,15) (3,13) (3,5) (4,12) (4,10) (4,13) (4,8) (4,5) (4,6) (4,16) (5,11) (5,14) (5,15) (5,13) (5,16) (5,9) (6,13) (6,9) (6,14) (6,11) (6,12) (7,16) (7,14) (7,15) (7,12) (7,11) (7,9) (7,13) (8,12) (8,10) (8,16) (8,14) (9,11) (9,13) (9,16) (9,10) (10,13) (10,15) (10,11) (11,15) (11,14) (12,14) (12,15) (13,15) (13,16) (13,14) (15,16)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,12,15,16,13,14,11,10,8,1,2,4,3,5,9,7", "difficulty": "hard", "doc_id": "270"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,1) (0,2) (0,10) (0,4) (1,9) (1,10) (1,6) (1,2) (1,5) (1,8) (1,11) (2,7) (2,3) (2,6) (2,4) (2,11) (2,10) (3,10) (3,5) (3,11) (4,7) (4,10) (4,8) (5,9) (5,8) (5,6) (6,7) (6,11) (7,9) (7,10) (7,8) (8,11) (9,11) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,8,11,10,7,9,5,6,1,2,3", "difficulty": "hard", "doc_id": "271"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,8) (0,12) (0,9) (0,1) (0,7) (0,11) (1,3) (1,11) (1,7) (1,14) (1,2) (2,11) (2,15) (2,6) (2,3) (2,13) (3,6) (3,13) (3,5) (3,9) (4,14) (4,12) (4,10) (4,6) (4,9) (4,8) (5,14) (5,9) (5,12) (5,10) (6,9) (6,14) (6,7) (6,8) (6,11) (7,11) (7,14) (7,9) (7,12) (7,13) (7,10) (7,8) (8,9) (8,10) (8,11) (9,12) (9,13) (10,13) (10,14) (10,11) (11,12) (12,15) (12,14) (12,13)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,12,13,10,14,7,8,6,4,9,5,3,1,2,15", "difficulty": "hard", "doc_id": "272"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 11, and the edges are: (0,11) (0,3) (0,9) (0,4) (0,2) (0,5) (0,8) (1,11) (1,6) (1,9) (1,8) (1,2) (1,4) (1,7) (2,11) (2,8) (2,6) (2,9) (2,3) (3,10) (3,11) (3,9) (3,8) (3,7) (3,6) (4,9) (4,11) (4,5) (4,6) (4,10) (4,7) (5,11) (5,9) (5,6) (5,7) (5,8) (6,10) (6,9) (6,7) (6,11) (7,8) (7,11) (8,11) (8,9) (8,10) (9,10) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,10,11,7,6,9,5,4,1,2,3", "difficulty": "hard", "doc_id": "273"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 10, and the edges are: (0,1) (0,9) (0,4) (0,6) (1,5) (1,6) (1,9) (1,8) (1,10) (2,3) (2,9) (2,10) (2,8) (2,7) (3,9) (3,4) (3,6) (4,8) (4,6) (4,5) (5,7) (5,6) (6,7) (6,8) (7,8) (7,9) (8,9) (9,10)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,8,9,10,1,5,7,2,3,4", "difficulty": "hard", "doc_id": "274"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (0,5) (0,1) (1,5) (1,6) (2,3) (3,6) (3,4) (4,5) (4,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,5,4,6,3,2", "difficulty": "easy", "doc_id": "275"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 8, and the edges are: (0,6) (0,3) (0,7) (0,1) (1,6) (1,4) (1,5) (2,5) (3,8) (4,7) (4,5) (4,6) (5,6) (5,7) (6,8) (7,8)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,3,8,7,4,6,1,5,2", "difficulty": "easy", "doc_id": "276"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 17, and the edges are: (0,13) (0,4) (0,1) (0,3) (0,2) (0,16) (0,15) (0,11) (0,8) (1,11) (1,2) (1,12) (1,3) (1,15) (1,17) (1,6) (1,8) (1,7) (1,16) (1,5) (2,15) (2,7) (2,11) (2,6) (2,12) (2,4) (2,13) (2,14) (2,16) (3,6) (3,16) (3,9) (3,17) (3,4) (3,5) (3,12) (3,8) (4,16) (4,15) (4,12) (4,8) (4,6) (5,13) (5,15) (5,10) (5,9) (5,11) (5,12) (5,16) (6,13) (6,17) (6,14) (6,12) (6,16) (6,7) (6,8) (7,13) (7,16) (7,8) (7,10) (7,17) (7,14) (7,12) (7,15) (7,11) (8,13) (8,17) (8,16) (8,12) (9,16) (9,11) (9,17) (9,13) (9,10) (10,13) (10,15) (10,14) (11,14) (12,15) (12,17) (12,13) (13,17) (13,15) (14,17) (14,16) (15,16) (16,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,8,12,13,15,16,17,14,11,9,10,7,6,4,3,5,1,2", "difficulty": "hard", "doc_id": "277"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,6) (0,8) (0,3) (0,9) (1,5) (1,8) (1,2) (1,3) (1,9) (1,4) (2,4) (2,8) (2,5) (3,8) (3,9) (4,6) (4,9) (4,5) (4,7) (4,8) (5,6) (6,7) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,8,4,7,6,5,2,1,3", "difficulty": "easy", "doc_id": "278"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,7) (0,8) (0,13) (0,2) (0,16) (1,2) (1,14) (1,7) (1,18) (1,9) (1,12) (1,3) (1,5) (1,17) (2,15) (2,4) (2,9) (2,12) (2,8) (2,16) (2,10) (3,5) (3,9) (3,15) (3,11) (3,8) (3,17) (4,15) (4,11) (4,5) (4,17) (4,6) (5,17) (5,14) (5,12) (5,16) (6,12) (6,7) (6,15) (6,17) (6,11) (7,15) (7,12) (8,10) (8,14) (8,9) (8,16) (8,15) (8,12) (9,13) (9,12) (9,14) (10,16) (10,12) (10,11) (10,18) (11,16) (12,17) (12,16) (13,15) (13,17) (14,16) (15,16) (15,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,16,15,18,10,11,6,17,13,9,14,8,12,7,1,3,5,4,2", "difficulty": "hard", "doc_id": "279"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,2) (1,2) (1,4) (1,3) (3,4)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,1,3,4", "difficulty": "easy", "doc_id": "280"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,8) (0,2) (0,1) (0,3) (0,16) (0,10) (0,17) (0,5) (0,9) (0,14) (0,7) (0,11) (1,11) (1,12) (1,2) (1,7) (1,13) (1,14) (1,16) (1,18) (1,5) (1,10) (1,6) (1,15) (1,17) (2,8) (2,11) (2,9) (2,3) (2,16) (2,13) (2,7) (2,5) (2,18) (2,4) (3,4) (3,13) (3,14) (3,16) (3,7) (3,15) (3,18) (4,11) (4,5) (4,15) (4,16) (4,7) (4,6) (4,9) (4,12) (4,13) (5,6) (5,15) (5,18) (5,13) (5,16) (5,7) (5,10) (5,17) (5,14) (5,9) (6,13) (6,12) (6,17) (6,10) (6,8) (6,16) (6,9) (6,7) (6,18) (7,17) (7,8) (7,9) (7,12) (7,14) (8,12) (8,18) (8,13) (8,15) (8,10) (9,18) (9,16) (9,10) (9,15) (9,14) (9,11) (10,17) (10,11) (10,16) (10,14) (10,15) (10,12) (11,14) (11,16) (11,15) (11,12) (11,17) (11,13) (12,18) (12,14) (12,15) (12,13) (13,18) (13,14) (13,17) (13,16) (14,17) (14,15) (15,16) (15,17) (16,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,11,13,16,18,12,15,17,14,10,9,7,8,6,5,4,3,2,1", "difficulty": "hard", "doc_id": "281"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 19, and the edges are: (0,8) (0,9) (0,18) (0,11) (0,15) (0,3) (0,19) (0,1) (1,13) (1,11) (1,15) (1,19) (1,2) (1,12) (1,4) (1,6) (1,9) (1,17) (2,10) (2,6) (2,12) (2,4) (2,13) (2,8) (2,5) (2,15) (3,13) (3,11) (3,5) (3,7) (3,18) (3,10) (4,8) (4,11) (4,9) (4,18) (4,13) (4,15) (4,12) (5,7) (5,9) (5,12) (5,16) (5,14) (5,18) (6,12) (6,7) (6,18) (6,14) (6,17) (6,10) (6,15) (6,8) (7,8) (7,16) (7,10) (7,11) (7,14) (8,19) (8,14) (8,17) (8,16) (8,11) (8,12) (9,12) (9,16) (9,13) (9,19) (10,16) (10,13) (10,18) (11,13) (11,15) (11,19) (12,19) (13,14) (13,18) (14,15) (14,18) (15,16) (15,19) (15,17) (16,17) (16,19) (17,19) (18,19)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,1,17,19,18,14,15,16,10,13,11,8,12,9,5,3,7,6,2,4", "difficulty": "hard", "doc_id": "282"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (0,7) (0,1) (0,4) (1,6) (1,2) (2,3) (3,6) (3,4) (3,7) (4,6) (5,7)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,6,1,2,3,7,5", "difficulty": "easy", "doc_id": "283"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,5) (0,13) (0,12) (0,1) (0,7) (0,4) (0,6) (1,8) (1,13) (1,7) (1,17) (1,9) (1,15) (1,12) (1,4) (2,3) (2,17) (2,16) (2,10) (2,4) (3,7) (3,8) (3,18) (3,6) (4,11) (4,5) (4,13) (4,18) (4,7) (5,16) (5,8) (5,18) (5,11) (5,7) (5,12) (5,15) (6,12) (6,17) (6,14) (7,11) (7,12) (7,13) (7,8) (7,9) (8,18) (8,11) (8,15) (8,9) (8,13) (9,18) (10,16) (10,11) (11,13) (11,16) (11,12) (12,15) (12,18) (14,15) (15,17)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,14,15,17,2,4,7,9,18,12,5,16,10,11,13,1,8,3", "difficulty": "hard", "doc_id": "284"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,5) (0,8) (0,7) (0,3) (0,6) (0,4) (0,1) (0,2) (1,4) (1,2) (1,9) (1,5) (2,8) (2,7) (2,5) (2,9) (2,6) (3,9) (3,4) (3,6) (3,8) (4,8) (4,6) (4,5) (5,8) (5,6) (6,8) (6,7) (7,8) (7,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,6,7,9,3,8,5,4,1", "difficulty": "easy", "doc_id": "285"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 15, and the edges are: (0,4) (0,3) (0,10) (1,9) (1,7) (1,11) (1,12) (1,13) (2,13) (2,15) (2,5) (3,12) (3,8) (3,14) (4,11) (4,8) (5,15) (5,7) (5,8) (5,13) (5,12) (6,13) (6,12) (7,12) (7,10) (7,8) (8,9) (9,12) (9,14) (10,15) (10,11) (12,15) (14,15)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,11,4,8,9,14,15,2,5,7,1,13,6,12,3", "difficulty": "hard", "doc_id": "286"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 18, and the edges are: (0,2) (0,6) (0,15) (0,1) (0,4) (0,10) (1,2) (1,16) (1,6) (1,12) (1,17) (1,5) (1,3) (1,18) (1,11) (1,8) (1,7) (2,4) (2,16) (2,14) (2,5) (2,10) (2,7) (2,12) (2,13) (2,9) (2,17) (3,12) (3,18) (3,15) (3,14) (3,6) (3,5) (3,16) (4,18) (4,15) (4,6) (5,18) (5,8) (5,11) (5,9) (5,10) (5,16) (5,15) (5,6) (5,12) (5,7) (6,13) (6,7) (6,17) (6,12) (6,14) (6,11) (6,9) (7,13) (7,15) (7,8) (7,11) (7,12) (7,14) (7,9) (8,10) (8,14) (8,18) (8,12) (8,16) (9,14) (9,18) (9,11) (9,16) (9,10) (9,13) (9,17) (9,15) (10,15) (10,16) (10,18) (10,11) (10,13) (10,17) (11,15) (11,16) (12,16) (12,17) (12,14) (12,15) (12,18) (13,14) (13,16) (14,15) (14,18) (14,17) (15,16) (15,18) (16,18) (17,18)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,10,17,18,16,15,14,13,9,11,7,12,8,5,6,4,2,1,3", "difficulty": "hard", "doc_id": "287"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 12, and the edges are: (0,12) (0,7) (0,6) (0,4) (0,3) (0,2) (0,8) (0,11) (0,9) (1,8) (1,11) (1,2) (1,5) (1,4) (1,12) (1,6) (2,8) (2,11) (2,3) (2,4) (2,7) (2,6) (3,9) (3,12) (3,7) (3,11) (3,10) (4,12) (4,9) (4,7) (4,5) (4,11) (5,10) (5,6) (6,12) (6,9) (7,11) (7,8) (7,9) (8,11) (9,10) (9,11) (9,12) (10,12) (10,11)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,9,12,10,11,8,7,4,5,6,1,2,3", "difficulty": "hard", "doc_id": "288"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 9, and the edges are: (0,7) (0,5) (0,4) (0,2) (1,9) (1,8) (1,3) (1,2) (1,4) (1,6) (2,8) (2,5) (2,4) (3,8) (3,4) (3,7) (3,5) (4,9) (4,6) (4,5) (5,7) (5,6) (6,9) (7,9) (7,8) (8,9)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,2,4,5,6,9,8,7,3,1", "difficulty": "easy", "doc_id": "289"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 14, and the edges are: (0,10) (0,11) (0,2) (0,12) (0,13) (0,5) (0,8) (0,9) (0,6) (1,10) (1,4) (1,7) (1,2) (1,9) (2,8) (2,7) (2,3) (2,9) (2,4) (2,10) (3,5) (3,8) (3,10) (3,14) (3,11) (3,7) (3,9) (3,6) (4,13) (4,5) (4,6) (4,10) (4,9) (4,12) (4,7) (5,6) (5,9) (5,10) (5,12) (5,14) (6,9) (6,14) (6,11) (6,10) (6,13) (7,14) (7,10) (7,13) (7,9) (7,8) (7,11) (8,11) (8,13) (8,12) (9,12) (9,11) (9,14) (10,14) (11,12) (11,13) (11,14) (12,13) (12,14) (13,14)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,6,13,14,12,11,9,7,8,3,10,5,4,2,1", "difficulty": "hard", "doc_id": "290"}
{"question": "In an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (4,2) (0,4) (4,3) (0,1) (0,2) (4,1) (2,3)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nTherefore, the path can be: 0,1,4,2,3.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,4) (1,5) (3,1) (4,2) (3,5) (2,1) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 1. As node 1 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 1,3,5,2,4,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 5, and the edges are: (0,2) (0,1) (4,5) (4,3) (4,2) (5,3) (1,4) (2,5)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 0. As node 0 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nThe path can be: 0,2,5,3,4,1.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 6, and the edges are: (1,4) (3,5) (3,6) (4,5) (0,3) (1,0) (2,4) (6,1)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 2. As node 2 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 3, and node 3 is not visited, we can then visit node 3.\nAs node 3 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nThe path can be: 2,4,5,3,6,1,0.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 7, and the edges are: (2,3) (5,1) (6,0) (5,3) (7,1) (3,7) (0,5) (6,5) (2,6) (3,4) (1,4) (2,0) (6,4) (1,6)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA: Yes. We can start at node 3. As node 3 is connected with node 4, and node 4 is not visited, we can then visit node 4.\nAs node 4 is connected with node 6, and node 6 is not visited, we can then visit node 6.\nAs node 6 is connected with node 2, and node 2 is not visited, we can then visit node 2.\nAs node 2 is connected with node 0, and node 0 is not visited, we can then visit node 0.\nAs node 0 is connected with node 5, and node 5 is not visited, we can then visit node 5.\nAs node 5 is connected with node 1, and node 1 is not visited, we can then visit node 1.\nAs node 1 is connected with node 7, and node 7 is not visited, we can then visit node 7.\nThe path can be: 3,4,6,2,0,5,1,7.\n\nIn an undirected graph, (i,j) means that node i and node j are connected with an undirected edge.\nThe nodes are numbered from 0 to 4, and the edges are: (0,2) (0,4) (0,1) (1,3) (1,2) (3,4)\nQ: Is there a path in this graph that visits every node exactly once? If yes, give the path. Note that in a path, adjacent nodes must be connected with edges.\nA:", "answer": "Yes. The path can be: 0,4,3,1,2", "difficulty": "easy", "doc_id": "291"}