| """Functions related to the Mycielski Operation and the Mycielskian family |
| of graphs. |
| |
| """ |
|
|
| import networkx as nx |
| from networkx.utils import not_implemented_for |
|
|
| __all__ = ["mycielskian", "mycielski_graph"] |
|
|
|
|
| @not_implemented_for("directed") |
| @not_implemented_for("multigraph") |
| @nx._dispatchable(returns_graph=True) |
| def mycielskian(G, iterations=1): |
| r"""Returns the Mycielskian of a simple, undirected graph G |
| |
| The Mycielskian of graph preserves a graph's triangle free |
| property while increasing the chromatic number by 1. |
| |
| The Mycielski Operation on a graph, :math:`G=(V, E)`, constructs a new |
| graph with :math:`2|V| + 1` nodes and :math:`3|E| + |V|` edges. |
| |
| The construction is as follows: |
| |
| Let :math:`V = {0, ..., n-1}`. Construct another vertex set |
| :math:`U = {n, ..., 2n}` and a vertex, `w`. |
| Construct a new graph, `M`, with vertices :math:`U \bigcup V \bigcup w`. |
| For edges, :math:`(u, v) \in E` add edges :math:`(u, v), (u, v + n)`, and |
| :math:`(u + n, v)` to M. Finally, for all vertices :math:`u \in U`, add |
| edge :math:`(u, w)` to M. |
| |
| The Mycielski Operation can be done multiple times by repeating the above |
| process iteratively. |
| |
| More information can be found at https://en.wikipedia.org/wiki/Mycielskian |
| |
| Parameters |
| ---------- |
| G : graph |
| A simple, undirected NetworkX graph |
| iterations : int |
| The number of iterations of the Mycielski operation to |
| perform on G. Defaults to 1. Must be a non-negative integer. |
| |
| Returns |
| ------- |
| M : graph |
| The Mycielskian of G after the specified number of iterations. |
| |
| Notes |
| ----- |
| Graph, node, and edge data are not necessarily propagated to the new graph. |
| |
| """ |
|
|
| M = nx.convert_node_labels_to_integers(G) |
|
|
| for i in range(iterations): |
| n = M.number_of_nodes() |
| M.add_nodes_from(range(n, 2 * n)) |
| old_edges = list(M.edges()) |
| M.add_edges_from((u, v + n) for u, v in old_edges) |
| M.add_edges_from((u + n, v) for u, v in old_edges) |
| M.add_node(2 * n) |
| M.add_edges_from((u + n, 2 * n) for u in range(n)) |
|
|
| return M |
|
|
|
|
| @nx._dispatchable(graphs=None, returns_graph=True) |
| def mycielski_graph(n): |
| """Generator for the n_th Mycielski Graph. |
| |
| The Mycielski family of graphs is an infinite set of graphs. |
| :math:`M_1` is the singleton graph, :math:`M_2` is two vertices with an |
| edge, and, for :math:`i > 2`, :math:`M_i` is the Mycielskian of |
| :math:`M_{i-1}`. |
| |
| More information can be found at |
| http://mathworld.wolfram.com/MycielskiGraph.html |
| |
| Parameters |
| ---------- |
| n : int |
| The desired Mycielski Graph. |
| |
| Returns |
| ------- |
| M : graph |
| The n_th Mycielski Graph |
| |
| Notes |
| ----- |
| The first graph in the Mycielski sequence is the singleton graph. |
| The Mycielskian of this graph is not the :math:`P_2` graph, but rather the |
| :math:`P_2` graph with an extra, isolated vertex. The second Mycielski |
| graph is the :math:`P_2` graph, so the first two are hard coded. |
| The remaining graphs are generated using the Mycielski operation. |
| |
| """ |
|
|
| if n < 1: |
| raise nx.NetworkXError("must satisfy n >= 1") |
|
|
| if n == 1: |
| return nx.empty_graph(1) |
|
|
| else: |
| return mycielskian(nx.path_graph(2), n - 2) |
|
|
