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Theorem 2.5 Every graph with average degree at least \( {2k} \), where \( k \) is a positive integer, has an induced subgraph with minimum degree at least \( k + 1 \) . | Proof Let \( G \) be a graph with average degree \( d\left( G\right) \geq {2k} \), and let \( F \) be an induced subgraph of \( G \) with the largest possible average degree and, subject to this, the smallest number of vertices. We show that \( \delta \left( F\right) \geq k + 1 \) . This is clearly true if \( v\left( F... | Yes |
Theorem 2.10 A graph \( G \) is even if and only if \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) . | Proof Suppose that \( \left| {\partial \left( X\right) }\right| \) is even for every subset \( X \) of \( V \) . Then, in particular, \( \left| {\partial \left( v\right) }\right| \) is even for every vertex \( v \) . But, as noted above, \( \partial \left( v\right) \) is just the set of all links incident with \( v \) ... | Yes |
Proposition 2.11 Let \( G \) be a graph, and let \( X \) and \( Y \) be subsets of \( V \) . Then\n\n\[ \partial \left( X\right) \bigtriangleup \partial \left( Y\right) = \partial \left( {X\bigtriangleup Y}\right) \] | Proof Consider the Venn diagram, shown in Figure 2.9, of the partition of \( V \)\n\n\[ \left( {X \cap Y,\;X \smallsetminus Y,\;Y \smallsetminus X,\;\bar{X} \cap \bar{Y}}\right) \]\n\ndetermined by the partitions \( \left( {X,\bar{X}}\right) \) and \( \left( {Y,\bar{Y}}\right) \), where \( \bar{X} \mathrel{\text{:=}} V... | Yes |
Theorem 2.15 In a connected graph \( G \), a nonempty edge cut \( \partial \left( X\right) \) is a bond if and only if both \( G\left\lbrack X\right\rbrack \) and \( G\left\lbrack {V \smallsetminus X}\right\rbrack \) are connected. | Proof Suppose, first, that \( \partial \left( X\right) \) is a bond, and let \( Y \) be a nonempty proper subset of \( X \) . Because \( G \) is connected, both \( \partial \left( Y\right) \) and \( \partial \left( {X \smallsetminus Y}\right) \) are nonempty. It follows that \( E\left\lbrack {Y, X \smallsetminus Y}\rig... | Yes |
Corollary 2.16 The symmetric difference of two even subgraphs is an even subgraph. | Proof Let \( {F}_{1} \) and \( {F}_{2} \) be even subgraphs of a graph \( G \), and let \( X \) be a subset of V. By Proposition 2.13,\n\n\[ \n{\partial }_{{F}_{1}\bigtriangleup {F}_{2}}\left( X\right) = {\partial }_{{F}_{1}}\left( X\right) \bigtriangleup {\partial }_{{F}_{2}}\left( X\right) \n\]\n\nBy Theorem 2.10, \(... | Yes |
Proposition 2.18 In any graph, every even subgraph meets every edge cut in an even number of edges. | Proof We first show that every cycle meets every edge cut in an even number of edges. Let \( C \) be a cycle and \( \partial \left( X\right) \) an edge cut. Each vertex of \( C \) is either in \( X \) or in \( V \smallsetminus X \) . As \( C \) is traversed, the number of times it crosses from \( X \) to \( V \smallset... | Yes |
For any two graphs \( F \) and \( G \) such that \( v\left( F\right) < v\left( G\right) \), the parameter \( \left( \begin{matrix} G \\ F \end{matrix}\right) \) is reconstructible. | Each copy of \( F \) in \( G \) occurs in exactly \( v\left( G\right) - v\left( F\right) \) of the vertex-deleted subgraphs \( G - v \) (namely, whenever the vertex \( v \) is not present in the copy).\n\nTherefore\n\[\n\left( \begin{array}{l} G \\ F \end{array}\right) = \frac{1}{v\left( G\right) - v\left( F\right) }\m... | Yes |
Corollary 2.21 For any two graphs \( F \) and \( G \) such that \( v\left( F\right) < v\left( G\right) \), the number of subgraphs of \( G \) that are isomorphic to \( F \) and include a given vertex \( v \) is reconstructible. | Proof This number is \( \left( \begin{matrix} G \\ F \end{matrix}\right) - \left( \begin{matrix} G - v \\ F \end{matrix}\right) \), which is reconstructible by Kelly’s Lemma. | Yes |
Corollary 2.22 The size and the degree sequence are reconstructible parameters. | Proof Take \( F = {K}_{2} \) in Kelly’s Lemma and Corollary 2.21, respectively. | Yes |
Theorem 2.25 The Möbius Inversion Formula\n\nLet \( f : {2}^{T} \rightarrow \mathbb{R} \) be a real-valued function defined on the subsets of a finite set \( T \) . Define the function \( g : {2}^{T} \rightarrow \mathbb{R} \) by\n\n\[ g\left( S\right) \mathrel{\text{:=}} \mathop{\sum }\limits_{{S \subseteq X \subseteq ... | Proof By the Binomial Theorem,\n\n\[ \mathop{\sum }\limits_{{S \subseteq X \subseteq Y}}{\left( -1\right) }^{\left| X\right| - \left| S\right| } = \mathop{\sum }\limits_{{\left| S\right| \leq \left| X\right| \leq \left| Y\right| }}\left( \begin{array}{l} \left| Y\right| - \left| S\right| \\ \left| X\right| - \left| S\r... | Yes |
Lemma 2.26 NASH-WILLIAMS' LEMMA\n\nLet \( G \) be a graph, \( F \) a spanning subgraph of \( G \), and \( H \) an edge reconstruction of \( G \) that is not isomorphic to \( G \) . Then\n\n\[ \n{\left| G \rightarrow G\right| }_{F} - {\left| G \rightarrow H\right| }_{F} = {\left( -1\right) }^{e\left( G\right) - e\left( ... | Proof By (2.6) and (2.7),\n\n\[ \n\mathop{\sum }\limits_{{F \subseteq X \subseteq G}}{\left| G \rightarrow H\right| }_{X} = \operatorname{aut}\left( F\right) \left( \begin{array}{l} H \\ F \end{array}\right) \n\]\n\nWe invert this identity by applying the Möbius Inversion Formula (identifying each spanning subgraph of ... | Yes |
Theorem 2.27 A graph \( G \) is edge reconstructible if there exists a spanning subgraph \( F \) of \( G \) such that either of the following two conditions holds.\n\n(i) \( {\left| G \rightarrow H\right| }_{F} \) takes the same value for all edge reconstructions \( H \) of \( G \) ,\n\n(ii) \( \left| {F \rightarrow G}... | Proof Let \( H \) be an edge reconstruction of \( G \) . If condition (i) holds, the left-hand side of (2.8) is zero whereas the right-hand side is nonzero. The inequality of condition (ii) is equivalent, by (2.6), to the inequality\n\n\[ \mathop{\sum }\limits_{{F \subseteq X \subseteq G}}{\left| G \rightarrow G\right|... | Yes |
Theorem 3.1 THE FRIENDSHIP THEOREM\n\nLet \( G \) be a simple graph in which any two vertices (people) have exactly one common neighbour (friend). Then \( G \) has a vertex of degree \( n - 1 \) (a politician, everyone's friend). | Proof Suppose the theorem false, and let \( G \) be a friendship graph with \( \Delta < n - 1 \) . Let us show first of all that \( G \) is regular. Consider two nonadjacent vertices \( x \) and \( y \), where, without loss of generality, \( d\left( x\right) \geq d\left( y\right) \) . By assumption, \( x \) and \( y \)... | Yes |
Theorem 3.4 If \( G \) is a connected even graph, the walk \( W \) returned by Fleury’s Algorithm is an Euler tour of \( G \) . | Proof The sequence \( W \) is initially a trail, and remains one throughout the procedure, because Fleury’s Algorithm always selects an edge of \( F \) (that is, an as yet unchosen edge) which is incident to the terminal vertex \( x \) of \( W \) . Moreover, the algorithm terminates when \( {\partial }_{F}\left( x\righ... | Yes |
Theorem 3.6 Let \( x \) and \( y \) be two vertices of a digraph \( D \) . Then \( y \) is reachable from \( x \) in \( D \) if and only if \( {\partial }^{ + }\left( X\right) \neq \varnothing \) for every subset \( X \) of \( V \) which contains \( x \) but not \( y \) . | Proof Suppose, first, that \( y \) is reachable from \( x \) by a directed path \( P \) . Consider any subset \( X \) of \( V \) which contains \( x \) but not \( y \) . Let \( u \) be the last vertex of \( P \) which belongs to \( X \) and let \( v \) be its successor on \( P \) . Then \( \left( {u, v}\right) \in {\pa... | Yes |
Proposition 3.8 If a graph has a cycle covering in which each edge is covered at most twice, then it has a cycle double cover. | Proof Let \( \mathcal{C} \) be a cycle covering of a graph \( G \) in which each edge is covered at most twice. The symmetric difference \( \bigtriangleup \{ E\left( C\right) \mid C \in \mathcal{C}\} \) of the edge sets of the cycles in \( \mathcal{C} \) is then the set of edges of \( G \) which are covered just once b... | Yes |
Proposition 4.2 Every nontrivial tree has at least two leaves. | If \( x \) is a leaf of a tree \( T \), the subgraph \( T - x \) is a tree with \( v\left( {T - x}\right) = v\left( T\right) - 1 \) and \( e\left( {T - x}\right) = e\left( T\right) - 1 \) . Because the trivial tree has no edges, we have, by induction on the number of vertices, the following relationship between the num... | No |
Theorem 4.5 Any tournament on \( {2k} \) vertices contains a copy of each branching on \( k + 1 \) vertices. | Proof Let \( {v}_{1},{v}_{2},\ldots ,{v}_{2k} \) be a median order of a tournament \( T \) on \( {2k} \) vertices, and let \( B \) be a branching on \( k + 1 \) vertices. Consider the intervals \( {v}_{1},{v}_{2},\ldots ,{v}_{i} \) , \( 1 \leq i \leq {2k} \) . We show, by induction on \( k \), that there is a copy of \... | Yes |
Theorem 4.7 A graph is bipartite if and only if it contains no odd cycle. | Proof Clearly, a graph is bipartite if and only if each of its components is bipartite, and contains an odd cycle if and only if one of its components contains an odd cycle. Thus, it suffices to prove the theorem for connected graphs.\n\nLet \( G\left\lbrack {X, Y}\right\rbrack \) be a connected bipartite graph. Then t... | Yes |
Proposition 4.9 Let \( G \) be a graph and e a link of \( G \) . Then\n\n\[ t\left( G\right) = t\left( {G \smallsetminus e}\right) + t\left( {G/e}\right) \] | ## Exercises\n\n\( \star \) 4.2.1 Let \( G \) be a connected graph and \( e \) a link of \( G \) .\na) Describe a one-to-one correspondence between the set of spanning trees of \( G \) that contain \( e \) and the set of spanning trees of \( G/e \) .\nb) Deduce Proposition 4.9. | No |
Theorem 4.10 Let \( T \) be a spanning tree of a connected graph \( G \), and let \( S \) be a subset of its cotree \( \bar{T} \). Then \( C \mathrel{\text{:=}} \bigtriangleup \left\{ {{C}_{e} : e \in S}\right\} \) is an even subgraph of \( G \). Moreover, \( C \cap \bar{T} = S \), and \( C \) is the only even subgraph... | Proof As each fundamental cycle \( {C}_{e} \) is an even subgraph, it follows from Corollary 2.16 that \( C \) is an even subgraph, too. Furthermore, \( C \cap \bar{T} = S \), because each edge of \( S \) appears in exactly one member of the family \( \left\{ {{C}_{e} : e \in S}\right\} \).\n\nLet \( {C}^{\prime } \) b... | Yes |
Corollary 4.11 Let \( T \) be a spanning tree of a connected graph \( G \) . Every even subgraph of \( G \) can be expressed uniquely as a symmetric difference of fundamental cycles with respect to \( T \) . | Proof Let \( C \) be an even subgraph of \( G \) and let \( S \mathrel{\text{:=}} C \cap \bar{T} \) . It follows from Theorem 4.10 that \( C = \bigtriangleup \left\{ {{C}_{e} : e \in S}\right\} \) and that this is the only way of expressing \( C \) as a symmetric difference of fundamental cycles with respect to \( T \)... | Yes |
Theorem 5.1 A connected graph on three or more vertices has no cut vertices if and only if any two distinct vertices are connected by two internally disjoint paths. | Proof Let \( G \) be a connected graph, and let \( v \) be a vertex of \( G \) . If any two vertices of \( G \) are connected by two internally disjoint paths, any two vertices of \( G - v \) are certainly connected by at least one path, so \( G - v \) is connected and \( v \) is not a cut vertex of \( G \) . This bein... | Yes |
Theorem 5.2 A connected graph is nonseparable if and only if any two of its edges lie on a common cycle. | Proof If \( G \) is separable, it may be decomposed into two nonempty connected subgraphs, \( {G}_{1} \) and \( {G}_{2} \), which have just one vertex \( v \) in common. Let \( {e}_{i} \) be an edge of \( {G}_{i} \) incident with \( v, i = 1,2 \) . If either \( {e}_{1} \) or \( {e}_{2} \) is a loop, there is clearly no... | Yes |
Proposition 5.3 Let \( G \) be a graph. Then:\n\na) any two blocks of \( G \) have at most one vertex in common,\n\nb) the blocks of \( G \) form a decomposition of \( G \) ,\n\nc) each cycle of \( G \) is contained in a block of \( G \) . | Proof (a) We establish the claim by contradiction. Suppose that there are distinct blocks \( {B}_{1} \) and \( {B}_{2} \) with at least two common vertices. Note that \( {B}_{1} \) and \( {B}_{2} \) are necessarily loopless. Because they are maximal nonseparable subgraphs of \( G \) , neither one contains the other, so... | Yes |
Theorem 5.4 The Splitting Lemma\n\nLet \( G \) be a nonseparable graph and let \( v \) be a vertex of \( G \) of degree at least four with at least two distinct neighbours. Then some two nonparallel edges incident to \( v \) can be split off so that the resulting graph is connected and has no cut edges. | Proof There are two graphs on three vertices and five edges which satisfy the hypotheses of the theorem, and it may be readily checked that the theorem holds for these two graphs. We proceed by induction on \( m \) . Let \( f \) be an edge of \( G \) not incident to \( v \), and set \( H \mathrel{\text{:=}} G \smallset... | No |
Theorem 5.5 The Cycle Double Conjecture is true if and only if it is true for all nonseparable cubic graphs. | Proof We have already noted that it suffices to prove the Cycle Double Cover Conjecture for nonseparable graphs. Consider such a graph \( G \) . By Veblen’s Theorem, we may assume that \( G \) has at least one vertex of odd degree. If \( G \) has a vertex \( v \) of degree two, with neighbours \( u \) and \( w \), let ... | Yes |
Proposition 5.6 Let \( F \) be a nontrivial proper subgraph of a nonseparable graph G. Then \( F \) has an ear in \( G \) . | Proof If \( F \) is a spanning subgraph of \( G \), the set \( E\left( G\right) \smallsetminus E\left( F\right) \) is nonempty because, by hypothesis, \( F \) is a proper subgraph of \( G \) . Any edge in \( E\left( G\right) \smallsetminus E\left( F\right) \) is then an ear of \( F \) in \( G \) . We may suppose, there... | Yes |
Theorem 5.10 Every connected graph without cut edges has a strong orientation. | Proof Let \( G \) be a connected graph without cut edges. By Proposition 5.9, it suffices to show that each block \( B \) of \( G \) has a strong orientation. We may assume that \( B \neq {K}_{1} \) . Moreover, because \( G \) has no cut edges, \( B \neq {K}_{2} \) . Thus \( B \) contains a cycle and, by Theorem 5.8, h... | Yes |
Proposition 5.11 Let \( F \) be a nontrivial proper nonseparable strong subdigraph of a nonseparable strong digraph \( D \) . Then \( F \) has a directed ear in \( D \) . | Proof Because \( D \) is nonseparable, \( F \) has an ear in \( D \), by Proposition 5.6. Among all such ears, we choose one in which the number of reverse arcs (those directed towards its initial vertex) is as small as possible. We show that this path \( {xPy} \) is in fact a directed ear.\n\nAssume the contrary, and ... | Yes |
Theorem 5.14 Every strong digraph admits a coherent feedback arc set. | Proof By induction on the number of arcs. Let \( D \) be a strong digraph. If \( D \) is a directed cycle, the statement is obviously true. If not, then by Theorem 5.13 there exists a proper strong subdigraph \( {D}^{\prime } \) and a directed ear \( {yPx} \) of \( {D}^{\prime } \) such that \( D = {D}^{\prime } \cup P... | No |
Theorem 5.15 Every strong digraph \( D \) has a strong spanning subdigraph with at most \( {2n} - 2 \) arcs. | Proof We may assume that \( D \) has no loops, deleting them if necessary. If \( D = {K}_{1} \) , the assertion is trivial. If not, we apply Theorem 5.13 to each block \( B \) of \( D \) . Consider a directed ear decomposition of \( B \) . Delete from \( B \) the arcs in directed ears of length one, thereby obtaining a... | Yes |
Theorem 6.2 Let \( T \) be a BFS-tree of a connected graph \( G \), with root \( r \). Then:\n\na) for every vertex \( v \) of \( G,\ell \left( v\right) = {d}_{T}\left( {r, v}\right) \), the level of \( v \) in \( T \), \n\nb) every edge of \( G \) joins vertices on the same or consecutive levels of \( T \); that is,\n... | Proof The proof of (a) is left to the reader (Exercise 6.1.1). To establish (b), it suffices to prove that if \( {uv} \in E \) and \( \ell \left( u\right) < \ell \left( v\right) \), then \( \ell \left( u\right) = \ell \left( v\right) - 1 \) .\n\nWe first establish, by induction on \( \ell \left( u\right) \), that if \(... | No |
Theorem 6.3 Let \( G \) be a connected graph. Then the values of the level function \( \ell \) returned by BFS are the distances in \( G \) from the root \( r \) :\n\n\[ \ell \left( v\right) = {d}_{G}\left( {r, v}\right) ,\;\text{ for all }v \in V \] | Proof By Theorem 6.2a, \( \ell \left( v\right) = {d}_{T}\left( {r, v}\right) \) . Moreover, \( {d}_{T}\left( {r, v}\right) \geq {d}_{G}\left( {r, v}\right) \) because \( T \) is a subgraph of \( G \) . Thus \( \ell \left( v\right) \geq {d}_{G}\left( {r, v}\right) \) . We establish the opposite inequality by induction o... | Yes |
Proposition 6.5 Let \( u \) and \( v \) be two vertices of \( G \), with \( f\left( u\right) < f\left( v\right) \).\n\na) If \( u \) and \( v \) are adjacent in \( G \), then \( l\left( v\right) < l\left( u\right) \).\n\nb) \( u \) is an ancestor of \( v \) in \( T \) if and only if \( l\left( v\right) < l\left( u\righ... | ## Proof\n\na) According to lines 8-12 of DFS, the vertex \( u \) is removed from the stack \( S \) only after all potential children (uncoloured neighbours) have been considered for addition to \( S \) . One of these neighbours is \( v \), because \( f\left( u\right) < f\left( v\right) \) . Thus \( v \) is added to th... | Yes |
Theorem 6.6 Let \( T \) be a DFS-tree of a graph \( G \) . Then every edge of \( G \) joins vertices which are related in \( T \) . | Proof This follows almost immediately from Proposition 6.5. Let \( {uv} \) be an edge of \( G \) . Without loss of generality, suppose that \( f\left( u\right) < f\left( v\right) \) . By Proposition 6.5a, \( l\left( v\right) < l\left( u\right) \) . Now Proposition 6.5b implies that \( u \) is an ancestor of \( v \), so... | Yes |
Problem 6.8 Minimum-Weight Spanning Tree\n\nGIVEN: a weighted connected graph \( G \) ,\n\nFIND: a minimum-weight spanning tree \( T \) in \( G \) . | ## The Jarník-Prim Algorithm\n\nThe Minimum-Weight Spanning Tree Problem (6.8) can be solved by means of a tree-search due to Jarník (1930) and Prim (1957). In this algorithm, which we call the Jarnik-Prim Algorithm, an arbitrary vertex \( r \) is selected as the root of \( T \), and at each stage the edge added to the... | Yes |
Theorem 6.10 Every Jarník-Prim tree is an optimal tree. | Proof Let \( T \) be a Jarník-Prim tree with root \( r \) . We prove, by induction on \( v\left( T\right) \) , that \( T \) is an optimal tree. The first edge added to \( T \) is an edge \( e \) of least weight in the edge cut associated with \( \{ r\} \) ; in other words, \( w\left( e\right) \leq w\left( f\right) \) f... | Yes |
Proposition 6.15 Let \( D \) be a directed graph, \( C \) a strong component of \( D \), and \( F \) a DFS-branching forest in D. Then \( F \cap C \) is a branching. | Proof Each component of \( F \cap C \) is contained in \( F \), and thus is a branching. Furthermore, vertices of \( C \) which are related in \( F \) necessarily belong to the same component of \( F \cap C \), because the directed path in \( F \) connecting them is contained in \( C \) also (Exercise 3.4.3).\n\nSuppos... | No |
Proposition 7.1 For any flow \( f \) in a network \( N\left( {x, y}\right) \) and any subset \( X \) of \( V \) such that \( x \in X \) and \( y \in V \smallsetminus X \) , \n\n\[ \n\operatorname{val}\left( f\right) = {f}^{ + }\left( X\right) - {f}^{ - }\left( X\right) \n\] | Proof From the definition of a flow and its value, we have \n\n\[ \n{f}^{ + }\left( v\right) - {f}^{ - }\left( v\right) = \left\{ \begin{matrix} \operatorname{val}\left( f\right) & \text{ if }v = x \\ 0 & \text{ if }v \in X \smallsetminus \{ x\} \end{matrix}\right. \n\] \n\nSumming these equations over \( X \) and simp... | No |
Theorem 7.3 For any flow \( f \) and any cut \( K \mathrel{\text{:=}} {\partial }^{ + }\left( X\right) \) in a network \( N \) ,\n\n\[\n\operatorname{val}\left( f\right) \leq \operatorname{cap}\left( K\right)\n\]\n\nFurthermore, equality holds in this inequality if and only if each arc in \( {\partial }^{ + }\left( X\r... | Proof By (7.2),\n\n\[{f}^{ + }\left( X\right) \leq {c}^{ + }\left( X\right) \text{ and }{f}^{ - }\left( X\right) \geq 0\]\n\n(7.4)\n\nThus, applying Proposition 7.1,\n\n\[\operatorname{val}\left( f\right) = {f}^{ + }\left( X\right) - {f}^{ - }\left( X\right) \leq {c}^{ + }\left( X\right) = \operatorname{cap}\left( K\ri... | Yes |
Corollary 7.4 Let \( f \) be a flow and \( K \) a cut. If \( \operatorname{val}\left( f\right) = \operatorname{cap}\left( K\right) \), then \( f \) is a maximum flow and \( K \) is a minimum cut. | Proof Let \( {f}^{ * } \) be a maximum flow and \( {K}^{ * } \) a minimum cut. By Theorem 7.3,\n\n\[ \operatorname{val}\left( f\right) \leq \operatorname{val}\left( {f}^{ * }\right) \leq \operatorname{cap}\left( {K}^{ * }\right) \leq \operatorname{cap}\left( K\right) \]\n\nBut, by hypothesis, \( \operatorname{val}\left... | Yes |
Proposition 7.5 Let \( f \) be a flow in a network \( N \) . If there is an \( f \) -incrementing path \( P \), then \( f \) is not a maximum flow. More precisely, the function \( {f}^{\prime } \) defined by (7.5) is a flow in \( N \) of value \( \operatorname{val}\left( {f}^{\prime }\right) = \operatorname{val}\left( ... | We refer to the flow \( {f}^{\prime } \) defined by (7.5) as the incremented flow based on \( P \) . Figure 7.3b shows the incremented flow in the network of Figure 7.3a based on the \( f \) -incrementing path \( x{v}_{1}{v}_{2}{v}_{3}y \) . | No |
Proposition 7.6 Let \( f \) be a flow in a network \( N \mathrel{\text{:=}} N\left( {x, y}\right) \) . Suppose that there is no \( f \) -incrementing path in \( N \) . Let \( X \) be the set of all vertices reachable from \( x \) by \( f \) -unsaturated paths, and set \( K \mathrel{\text{:=}} {\partial }^{ + }\left( X\... | Proof Clearly \( x \in X \) . Also, \( y \in V \smallsetminus X \) because there is no \( f \) -incrementing path. Therefore \( K \) is a cut in \( N \) . Consider an arc \( a \in {\partial }^{ + }\left( X\right) \), with tail \( u \) and head \( v \) . Because \( u \in X \), there exists an \( f \) -unsaturated \( \le... | Yes |
Theorem 7.7 The Max-Flow Min-Cut Theorem\n\nIn any network, the value of a maximum flow is equal to the capacity of a minimum cut. | Proof Let \( f \) be a maximum flow. By Proposition 7.5, there can be no \( f \) - incrementing path. The theorem now follows from Proposition 7.6. | No |
Theorem 7.8 A flow \( f \) in a network is a maximum flow if and only if there is no \( f \) -incrementing path. | This theorem is the basis of an algorithm for finding a maximum flow in a network. Starting with a known flow \( f \), for instance the zero flow, we search for an \( f \) -incrementing path by means of a tree-search algorithm. An \( x \) -tree \( T \) is \( f \) - unsaturated if, for every vertex \( v \) of \( T \), t... | Yes |
Lemma 7.12 Let \( f \) be a nonzero circulation in a digraph. Then the support of \( f \) contains a cycle. Moreover, if \( f \) is nonnegative, then the support of \( f \) contains a directed cycle. | Proof The first assertion follows directly from Theorem 2.1, because the support of a nonzero circulation can contain no vertex of degree less than two. Likewise, the second assertion follows from Exercise 2.1.11a. | No |
Proposition 7.13 Every circulation in a digraph is a linear combination of the circulations associated with its cycles. | Proof Let \( f \) be a circulation, with support \( S \) . We proceed by induction on \( \left| S\right| \) . There is nothing to prove if \( S = \varnothing \) . If \( S \) is nonempty, then \( S \) contains a cycle \( C \) by Lemma 7.12. Let \( a \) be any arc of \( C \), and choose the sense of traversal of \( C \) ... | Yes |
Problem 8.4 Maximum-Weight Spanning Tree\n\nGIVEN: a weighted connected graph \( G \) ,\n\nFIND: a maximum-weight spanning tree in \( G \) . | In order to solve an instance of this problem, it suffices to replace each weight by its negative and apply the Jarník-Prim Algorithm (6.9) to find an optimal tree in the resulting weighted graph. The very same tree will be one of maximum weight in the original weighted graph. (We remark that one can similarly reduce t... | Yes |
Problem 8.5 Internally Disjoint Directed Paths (IDDP)\n\nGIVEN: a digraph \( D \mathrel{\text{:=}} D\left( {x, y}\right) \) ,\n\nFIND: a maximum family of internally disjoint directed \( \left( {x, y}\right) \) -paths in \( D \) . | A polynomial reduction of IDDP to ADDP can be obtained by constructing a new digraph \( {D}^{\prime } \mathrel{\text{:=}} {D}^{\prime }\left( {x, y}\right) \) from \( D \) as follows.\n\n\( \vartriangleright \) Split each vertex \( v \in V \smallsetminus \{ x, y\} \) into two new vertices \( {v}^{ - } \) and \( {v}^{ +... | Yes |
Problem 8.6 Disjoint Paths\n\nGIVEN: a graph \( G \), a positive integer \( k \), and two \( k \) -subsets \( X \) and \( Y \) of \( V \), \n\nDECIDE: Does \( G \) have \( k \) disjoint \( \left( {X, Y}\right) \) -paths? | Describe a polynomial reduction of this problem to IDP (INTERNALLY DISJOINT PATHS). | No |
Problem 8.7 Boolean Satisfiability (Sat)\n\nGIVEN: a boolean formula \( f \) ,\n\nDECIDE: Is \( f \) satisfiable? | Observe that SAT belongs to \( \mathcal{{NP}} \) : given appropriate values of the variables, it can be checked in polynomial time that the value of the formula is indeed 1 . These values of the variables therefore constitute a succinct certificate. Cook (1971) and Levin (1973) proved, independently, that SAT is an exa... | No |
The problem SAT is \( \mathcal{{NP}} \) -complete. | The proof of the Cook-Levin Theorem involves the notion of a Turing machine, and is beyond the scope of this book. A proof may be found in Garey and Johnson (1979) or Sipser (2005). | No |
Theorem 8.10 The problem 3-SAT is \( \mathcal{{NP}} \) -complete. | Proof By the Cook-Levin Theorem (8.8), it suffices to prove that SAT \( \preccurlyeq \) 3-SAT. Let \( f \) be a boolean formula in conjunctive normal form. We show how to construct, in polynomial time, a boolean formula \( {f}^{\prime } \) in conjunctive normal form such that:\ni) each clause in \( {f}^{\prime } \) has... | Yes |
Theorem 8.13 3-SAT \( \preccurlyeq \) EXACT COVER. | Proof Let \( f \) be an instance of 3-SAT, with clauses \( {f}_{1},\ldots ,{f}_{n} \) and variables \( {x}_{1},\ldots ,{x}_{m} \) . The first step is to construct a graph \( G \) from \( f \), by setting:\n\n\[ V\left( G\right) \mathrel{\text{:=}} \left\{ {{x}_{i} : 1 \leq i \leq m}\right\} \cup \left\{ {\overline{{x}_... | No |
Problem 8.16 Maximum Clique (Max Clique)\n\nGIVEN: a graph \( G \) ,\n\nFIND: a maximum clique in \( G \) . | In order to solve this problem, one needs to know, for a given value of \( k \), whether \( G \) has a \( k \) -clique. The largest such \( k \) is called the clique number of \( G \), denoted \( \omega \left( G\right) \) . If \( k \) is a fixed integer not depending on \( n \), the existence of a \( k \) -clique can b... | No |
Problem 8.19 Maximum Cut (Max Cut)\n\nGIVEN: a weighted graph \( \left( {G, w}\right) \) ,\n\nFIND: a maximum-weight spanning bipartite subgraph \( F \) of \( G \) . | This problem admits a polynomial-time 2-approximation algorithm, based on the ideas for the unweighted case presented in Chapter 2 (Exercise 2.2.2). We leave the details as an exercise (8.4.1). | No |
Problem 8.20 Metric Travelling Salesman Problem (Metric TSP) GIVEN: a weighted complete graph \( G \) whose weights satisfy inequality (8.3), FIND: a minimum-weight Hamilton cycle \( C \) of \( G \) . | Theorem 8.21 METRIC TSP | No |
Theorem 8.21 METRIC TSP admits a polynomial-time 2-approximation algorithm. | Proof Applying the Jarnik-Prim Algorithm (6.9), we first find a minimum-weight spanning tree \( T \) of \( G \) . Suppose that \( C \) is a minimum-weight Hamilton cycle of \( G \) . By deleting any edge of \( C \), we obtain a Hamilton path \( P \) of \( G \) . Because \( P \) is a spanning tree of \( G \) and \( T \)... | Yes |
Proposition 8.25 Weak Duality Theorem\n\nLet \( \mathbf{x} \) be a feasible solution to (8.4) and \( \mathbf{y} \) a feasible solution to its dual (8.5). Then\n\n\[ \mathrm{{cx}} \leq \mathrm{{yb}} \] | Proof Because \( \mathbf{c} \leq \mathbf{{yA}} \) and \( \mathbf{x} \geq \mathbf{0} \), we have \( \mathbf{{cx}} \leq \mathbf{{yAx}} \) . Likewise \( \mathbf{{yAx}} \leq \mathbf{{yb}} \) . Inequality (8.6) follows. | Yes |
Theorem 8.28 Suppose that \( \mathbf{A} \) is a totally unimodular matrix and that \( \mathbf{b} \) is an integer vector. If (8.4) has an optimal solution, then it has an integer optimal solution. | Proof The set of points in \( {\mathbb{R}}^{n} \) at which any single constraint holds with equality is a hyperplane in \( {\mathbb{R}}^{n} \) . Thus each constraint is satisfied by the points of a closed half-space of \( {\mathbb{R}}^{n} \), and the set of feasible solutions is the intersection of all these half-space... | Yes |
Theorem 8.31 When \( G \) is bipartite,(8.12) and (8.13) have \( \left( {0,1}\right) \) -valued optimal solutions. | If \( \mathbf{x} \) is a \( \left( {0,1}\right) \) -valued feasible solution to (8.12), then no two edges of the set \( M \mathrel{\text{:=}} \left\{ {e \in E : {x}_{e} = 1}\right\} \) have an end in common; that is, \( M \) is a matching of \( G \) . Analogously, if \( \mathbf{y} \) is a \( \left( {0,1}\right) \) -val... | Yes |
Theorem 9.2 If \( G \) has at least one pair of nonadjacent vertices,\n\n\[ \kappa \left( G\right) = \min \{ p\left( {u, v}\right) : u, v \in V, u \neq v,{uv} \notin E\} \]\n\n(9.3) | Proof If \( G \) has an edge \( e \) which is either a loop or one of a set of parallel edges, we can establish the theorem by deleting \( e \) and applying induction. So we may assume that \( G \) is simple.\n\nBy (9.1), \( \kappa \left( G\right) = \min \{ p\left( {u, v}\right) : u, v \in V, u \neq v\} \) . Let this m... | Yes |
Lemma 9.3 Let \( G \) be a \( k \) -connected graph and let \( H \) be a graph obtained from \( G \) by adding a new vertex \( y \) and joining it to at least \( k \) vertices of \( G \) . Then \( H \) is also \( k \) -connected. | Proof The conclusion clearly holds if any two vertices of \( H \) are adjacent, because \( v\left( H\right) \geq k + 1 \) . Let \( S \) be a subset of \( V\left( H\right) \) with \( \left| S\right| = k - 1 \) . To complete the proof, it suffices to show that \( H - S \) is connected.\n\nSuppose first that \( y \in S \)... | Yes |
Proposition 9.4 Let \( G \) be a \( k \) -connected graph, and let \( X \) and \( Y \) be subsets of \( V \) of cardinality at least \( k \) . Then there exists in \( G \) a family of \( k \) pairwise disjoint \( \left( {X, Y}\right) \) -paths. | Proof Obtain a new graph \( H \) from \( G \) by adding vertices \( x \) and \( y \) and joining \( x \) to each vertex of \( X \) and \( y \) to each vertex of \( Y \) . By Lemma 9.3, \( H \) is \( k \) - connected. Therefore, by Menger’s Theorem, there exist \( k \) internally disjoint \( {xy} \) - paths in \( H \) .... | Yes |
Theorem 9.6 Let \( S \) be a set of \( k \) vertices in a \( k \) -connected graph \( G \), where \( k \geq 2 \) . Then there is a cycle in \( G \) which includes all the vertices of \( S \) . | Proof By induction on \( k \) . We have already observed that the assertion holds for \( k = 2 \), so assume that \( k \geq 3 \) . Let \( x \in S \), and set \( T \mathrel{\text{:=}} S \smallsetminus x \) . Because \( G \) is \( k \) -connected, it is \( \left( {k - 1}\right) \) -connected. Therefore, by the induction ... | Yes |
Theorem 9.7 Menger's Theorem (Edge Version)\n\nFor any graph \( G\left( {x, y}\right) \) ,\n\n\[ {p}^{\prime }\left( {x, y}\right) = {c}^{\prime }\left( {x, y}\right) \] | This theorem was proved in Chapter 7 using flows. It may also be deduced from Theorem 9.1 by considering a suitable line graph (see Exercise 9.3.11). | No |
Theorem 9.9 Let \( G \) be a 2-connected graph and let \( S \) be a 2-vertex cut of \( G \) . Then the marked \( S \) -components of \( G \) are also 2-connected. | Proof Let \( H \) be a marked \( S \) -component of \( G \), with vertex set \( S \cup X \) . Then \( \left| {V\left( H\right) }\right| = \left| S\right| + \left| X\right| \geq 3 \) . Thus if \( H \) is complete, it is 2-connected. On the other hand, if \( H \) is not complete, every vertex cut of \( H \) is also a ver... | Yes |
Lemma 9.11 Let \( G \) be a 3-connected graph on at least five vertices, and let \( e = {xy} \) be an edge of \( G \) such that \( G/e \) is not 3-connected. Then there exists a vertex \( z \) such that \( \{ x, y, z\} \) is a 3-vertex cut of \( G \) . | Proof Let \( \{ z, w\} \) be a 2-vertex cut of \( G/e \) . At least one of these two vertices, say \( z \), is not the vertex resulting from the contraction of \( e \) . Set \( F \mathrel{\text{:=}} G - z \) .\n\nBecause \( G \) is 3-connected, \( F \) is certainly 2-connected. However \( F/e = \left( {G - z}\right) /e... | No |
Theorem 9.12 Let \( G \) be a 3-connected graph, let \( v \) be a vertex of \( G \) of degree at least four, and let \( H \) be an expansion of \( G \) at \( v \) . Then \( H \) is \( 3 \) -connected. | Proof Because \( G - v \) is 2-connected and \( {v}_{1} \) and \( {v}_{2} \) each have at least two neighbours in \( G - v \), the graph \( H \smallsetminus e \) is 2-connected, by Lemma 9.3. Using the fact that any two vertices of \( G \) are connected by three internally disjoint paths, it is now easily seen that any... | No |
Proposition 9.13 The atoms of a graph are pairwise disjoint. | Proof Let \( X \) and \( Y \) be two distinct atoms of a graph \( G \) . Suppose that \( X \cap Y \neq \varnothing \) . Because \( X \) and \( Y \) are atoms, neither is properly contained in the other, so \( X \cap \bar{Y} \) and \( \bar{X} \cap Y \) are both nonempty. We show that \( \bar{X} \cap \bar{Y} \) is nonemp... | Yes |
Theorem 9.14 Let \( G \) be a simple connected vertex-transitive graph of positive degree \( d \) . Then \( {\kappa }^{\prime } = d \) . | Proof Let \( X \) be an atom of \( G \), and let \( u \) and \( v \) be two vertices in \( X \) . Because \( G \) is vertex-transitive, it has an automorphism \( \theta \) such that \( \theta \left( u\right) = v \) . Being the image of an atom under an automorphism, the set \( \theta \left( X\right) \) is also an atom ... | Yes |
Theorem 9.16 Let \( G \) be a graph which is locally \( {2k} \) -edge-connected modulo \( v \) , where \( v \) is a vertex of even degree in \( G \) . Given any link uv incident with \( v \), there exists a second link vw incident with \( v \) such that the graph \( {G}^{\prime } \) obtained by splitting off \( {uv} \)... | Proof We may assume that \( n \geq 3 \) as the statement holds trivially when \( n = 2 \) . We may also assume that \( G \) is loopless. Consider all nonempty proper subsets \( X \) of \( V \smallsetminus \{ v\} \) . Splitting off \( {uv} \) and another link \( {vw} \) incident with \( v \) preserves the degree of \( X... | Yes |
Proposition 9.18 Let \( \left( {T, w}\right) \) be a Gomory-Hu tree of a graph \( G \) . For any two vertices \( x \) and \( y \) of \( G,{c}^{\prime }\left( {x, y}\right) \) is the minimum of the weights of the edges on the unique \( {xy} \) -path in \( T \) . | Proof Clearly, for every edge \( e \) on the \( {xy} \) -path in \( T \), the edge cut \( {B}_{e} \) associated with \( e \) separates \( x \) and \( y \) . If \( {v}_{1},{v}_{2},\ldots ,{v}_{k} \) is the \( {xy} \) -path in \( T \), where \( x = {v}_{1} \) and \( y = {v}_{k} \), it follows that\n\n\[ \n{c}^{\prime }\l... | No |
Theorem 9.19 Let \( G \) be a connected chordal graph which is not complete, and let \( S \) be a minimal vertex cut of \( G \) . Then \( S \) is a clique cut of \( G \) . | Proof Suppose that \( S \) contains two nonadjacent vertices \( x \) and \( y \) . Let \( {G}_{1} \) and \( {G}_{2} \) be two components of \( G - S \) . Because \( S \) is a minimal cut, both \( x \) and \( y \) are joined to vertices in both \( {G}_{1} \) and \( {G}_{2} \) . Let \( {P}_{i} \) be a shortest \( {xy} \)... | Yes |
Theorem 9.20 Let \( G \) be a connected chordal graph, and let \( {V}_{1} \) be a maximal clique of \( G \) . Then the maximal cliques of \( G \) can be arranged in a sequence \( \left( {{V}_{1},{V}_{2},\ldots ,{V}_{k}}\right) \) such that \( {V}_{j} \cap \left( {{ \cup }_{i = 1}^{j - 1}{V}_{i}}\right) \) is a clique o... | Proof There is nothing to prove if \( G \) is complete, so we may assume that \( G \) has a minimal vertex cut \( S \) . By Theorem 9.19, \( S \) is a clique of \( G \) . Let \( {H}_{i},1 \leq i \leq p \) , be the \( S \) -components of \( G \), and let \( {Y}_{i} \) be a maximal clique of \( {H}_{i} \) containing \( S... | No |
Theorem 9.21 Every chordal graph which is not complete has two nonadjacent simplicial vertices. | Proof Let \( \left( {{V}_{1},{V}_{2},\ldots ,{V}_{k}}\right) \) be a simplicial decomposition of a chordal graph, and let \( x \in {V}_{k} \smallsetminus \left( {{ \cup }_{i = 1}^{k - 1}{V}_{i}}\right) \) . Then \( x \) is a simplicial vertex. Now consider a simplicial decomposition \( \left( {{V}_{\pi \left( 1\right) ... | Yes |
Theorem 9.23 A graph is chordal if and only if it is the intersection graph of a family of subtrees of a tree. | Proof Let \( G \) be a chordal graph. By Theorem 9.20, \( G \) has a simplicial decomposition \( \left( {{V}_{1},{V}_{2},\ldots ,{V}_{k}}\right) \) . We prove by induction on \( k \) that \( G \) is the intersection graph of a family of subtrees \( \mathcal{T} = \left\{ {{T}_{v} : v \in V}\right\} \) of a tree \( T \) ... | Yes |
Theorem 10.1 The Jordan Curve Theorem\n\nAny simple closed curve \( C \) in the plane partitions the rest of the plane into two disjoint arcwise-connected open sets. | Although this theorem is intuitively obvious, giving a formal proof of it is quite tricky. The two open sets into which a simple closed curve \( C \) partitions the plane are called the interior and the exterior of \( C \) . We denote them by \( \operatorname{int}\left( C\right) \) and \( \operatorname{ext}\left( C\rig... | No |
Theorem 10.2 \( {K}_{5} \) is nonplanar. | Proof By contradiction. Let \( G \) be a planar embedding of \( {K}_{5} \), with vertices \( {v}_{1},{v}_{2},{v}_{3},{v}_{4},{v}_{5} \) . Because \( G \) is complete, any two of its vertices are joined by an edge. Now the cycle \( C \mathrel{\text{:=}} {v}_{1}{v}_{2}{v}_{3}{v}_{1} \) is a simple closed curve in the pla... | Yes |
Theorem 10.4 A graph \( G \) is embeddable on the plane if and only if it is embeddable on the sphere. | Proof Suppose that \( G \) has an embedding \( \widetilde{G} \) on the sphere. Choose a point \( z \) of the sphere not in \( \widetilde{G} \) . Then the image of \( \widetilde{G} \) under stereographic projection from \( z \) is an embedding of \( G \) on the plane. The converse is proved similarly. | Yes |
Proposition 10.5 Let \( G \) be a planar graph, and let \( f \) be a face in some planar embedding of \( G \) . Then \( G \) admits a planar embedding whose outer face has the same boundary as \( f \) . | Proof Consider an embedding \( \widetilde{G} \) of \( G \) on the sphere; such an embedding exists by virtue of Theorem 10.4. Denote by \( \widetilde{f} \) the face of \( \widetilde{G} \) corresponding to \( f \) . Let \( z \) be a point in the interior of \( \widetilde{f} \), and let \( \pi \left( \widetilde{G}\right)... | Yes |
Theorem 10.7 In a nonseparable plane graph other than \( {K}_{1} \) or \( {K}_{2} \), each face is bounded by a cycle. | Proof Let \( G \) be a nonseparable plane graph. Consider an ear decomposition \( {G}_{0},{G}_{1},\ldots ,{G}_{k} \) of \( G \), where \( {G}_{0} \) is a cycle, \( {G}_{k} = G \), and, for \( 0 \leq i \leq k - 2 \) , \( {G}_{i + 1} \mathrel{\text{:=}} {G}_{i} \cup {P}_{i} \) is a nonseparable plane subgraph of \( G \),... | Yes |
Corollary 10.8 In a loopless 3-connected plane graph, the neighbours of any vertex lie on a common cycle. | Proof Let \( G \) be a loopless 3-connected plane graph and let \( v \) be a vertex of \( G \) . Then \( G - v \) is nonseparable, so each face of \( G - v \) is bounded by a cycle, by Theorem 10.7. If \( f \) is the face of \( G - v \) in which the vertex \( v \) was situated, the neighbours of \( v \) lie on its boun... | Yes |
Theorem 10.10 If \( G \) is a plane graph, \[ \mathop{\sum }\limits_{{f \in F}}d\left( f\right) = {2m} \] | Proof Let \( {G}^{ * } \) be the dual of \( G \) . By (10.1) and Theorem 1.1, \[ \mathop{\sum }\limits_{{f \in F\left( G\right) }}d\left( f\right) = \mathop{\sum }\limits_{{{f}^{ * } \in V\left( {G}^{ * }\right) }}d\left( {f}^{ * }\right) = {2e}\left( {G}^{ * }\right) = {2e}\left( G\right) = {2m} \] | Yes |
Proposition 10.12 Let \( G \) be a connected plane graph, and let \( e \) be an edge of \( G \) that is not a cut edge. Then \[ {\left( G \smallsetminus e\right) }^{ * } \cong {G}^{ * }/{e}^{ * } \] | Proof Because \( e \) is not a cut edge, the two faces of \( G \) incident with \( e \) are distinct; denote them by \( {f}_{1} \) and \( {f}_{2} \) . Deleting \( e \) from \( G \) results in the amalgamation of \( {f}_{1} \) and \( {f}_{2} \) into a single face \( f \) (see Figure 10.12). Any face of \( G \) that is a... | Yes |
Proposition 10.13 Let \( G \) be a connected plane graph, and let \( e \) be a link of \( G \) . Then\n\n\[{\left( G/e\right) }^{ * } \cong {G}^{ * } \smallsetminus {e}^{ * }\] | Proof Because \( G \) is connected, \( {G}^{* * } \cong G \) (Exercise 10.2.4). Also, because \( e \) is not a loop of \( G \), the edge \( {e}^{ * } \) is not a cut edge of \( {G}^{ * } \), so \( {G}^{ * } \smallsetminus {e}^{ * } \) is connected. By Proposition 10.12,\n\n\[{\left( {G}^{ * } \smallsetminus {e}^{ * }\r... | No |
Theorem 10.14 The dual of a nonseparable plane graph is nonseparable. | Proof By induction on the number of edges. Let \( G \) be a nonseparable plane graph. The theorem is clearly true if \( G \) has at most one edge, so we may assume that \( G \) has at least two edges, hence no loops or cut edges. Let \( e \) be an edge of \( G \) . Then either \( G \smallsetminus e \) or \( G/e \) is n... | No |
Theorem 10.16 Let \( G \) be a connected plane graph, and let \( {G}^{ * } \) be a plane dual of \( G \). a) If \( C \) is a cycle of \( G \), then \( {C}^{ * } \) is a bond of \( {G}^{ * } \). | Proof a) Let \( C \) be a cycle of \( G \), and let \( {X}^{ * } \) denote the set of vertices of \( {G}^{ * } \) that lie in \( \operatorname{int}\left( C\right) \) . Then \( {C}^{ * } \) is the edge cut \( \partial \left( {X}^{ * }\right) \) in \( {G}^{ * } \) . By Proposition 10.15, the subgraph of \( {G}^{ * } \) i... | Yes |
Theorem 10.19 Euler's Formula\n\nFor a connected plane graph \( G \) ,\n\n\[ v\left( G\right) - e\left( G\right) + f\left( G\right) = 2 \] | Proof By induction on \( f\left( G\right) \), the number of faces of \( G \) . If \( f\left( G\right) = 1 \), each edge of \( G \) is a cut edge and so \( G \), being connected, is a tree. In this case \( e\left( G\right) = v\left( G\right) - 1 \) , by Theorem 4.3, and the assertion holds. Suppose that it is true for a... | Yes |
Corollary 10.20 All planar embeddings of a connected planar graph have the same number of faces. | Proof Let \( \widetilde{G} \) be a planar embedding of a planar graph \( G \) . By Euler’s Formula (10.2), we have\n\n\[ f\left( \widetilde{G}\right) = e\left( \widetilde{G}\right) - v\left( \widetilde{G}\right) + 2 = e\left( G\right) - v\left( G\right) + 2 \]\n\nThus the number of faces of \( \widetilde{G} \) depends ... | Yes |
Corollary 10.21 Let \( G \) be a simple planar graph on at least three vertices. Then \( m \leq {3n} - 6 \) . Furthermore, \( m = {3n} - 6 \) if and only if every planar embedding of \( G \) is a triangulation. | Proof It clearly suffices to prove the corollary for connected graphs. Let \( G \) be a simple connected planar graph with \( n \geq 3 \) . Consider any planar embedding \( \widetilde{G} \) of \( G \) . Because \( G \) is simple and connected, on at least three vertices, \( d\left( f\right) \geq 3 \) for all \( f \in F... | Yes |
Corollary 10.22 Every simple planar graph has a vertex of degree at most five. | Proof This is trivial for \( n < 3 \) . If \( n \geq 3 \), then by Theorem 1.1 and Corollary 10.21,\n\n\[ \n{\delta n} \leq \mathop{\sum }\limits_{{v \in V}}d\left( v\right) = {2m} \leq {6n} - {12} \n\]\n\nIt follows that \( \delta \leq 5 \) . | No |
Theorem 10.25 Overlapping bridges are either skew or else equivalent 3-bridges. | Proof Suppose that bridges \( B \) and \( {B}^{\prime } \) overlap. Clearly, each must have at least two vertices of attachment. If either \( B \) or \( {B}^{\prime } \) is a 2-bridge, it is easily verified that they must be skew. We may therefore assume that both \( B \) and \( {B}^{\prime } \) have at least three ver... | Yes |
Theorem 10.26 Inner (outer) bridges avoid one another. | Proof Let \( B \) and \( {B}^{\prime } \) be inner bridges of a cycle \( C \) in a plane graph \( G \) . Suppose that they overlap. By Theorem 10.25, they are either skew or equivalent 3-bridges. In both cases, we obtain contradictions.\n\nCase 1: \( B \) and \( {B}^{\prime } \) are skew. By definition, there exist dis... | Yes |
Theorem 10.27 A cycle in a simple 3-connected plane graph is a facial cycle if and only if it is nonseparating. | Proof Let \( G \) be a simple 3-connected plane graph and let \( C \) be a cycle of \( G \) . Suppose, first, that \( C \) is not a facial cycle of \( G \) . Then \( C \) has at least one inner bridge and at least one outer bridge. Because \( G \) is simple and connected, these bridges are not loops. Thus either they a... | Yes |
Theorem 10.28 Every simple 3-connected planar graph has a unique planar embedding. | Proof Let \( G \) be a simple 3-connected planar graph. By Theorem 10.27, the facial cycles in any planar embedding of \( G \) are precisely its nonseparating cycles. Because the latter are defined solely in terms of the abstract structure of the graph, they are the same for every planar embedding of \( G \) . | Yes |
A graph is planar if and only if it has no Kuratowski minor. | We remarked above that a graph which contains an \( F \) -subdivision also has an \( F \) -minor. Thus Kuratowski’s Theorem implies Wagner’s Theorem. On the other hand, because \( {K}_{3,3} \) has maximum degree three, any graph which has a \( {K}_{3,3} \) -minor contains a \( {K}_{3,3} \) -subdivision (Exercise 10.5.3... | No |
Lemma 10.33 Let \( G \) be a graph with a 2-vertex cut \( \{ x, y\} \) . Then each marked \( \{ x, y\} \) -component of \( G \) is isomorphic to a minor of \( G \) . | Proof Let \( H \) be an \( \{ x, y\} \) -component of \( G \), with marker edge \( e \), and let \( {xPy} \) be a path in another \( \{ x, y\} \) -component of \( G \) . Then \( H \cup P \) is a subgraph of \( G \) . But \( H \cup P \) is isomorphic to a subdivision of \( G + e \), so \( G + e \) is isomorphic to a min... | No |
Lemma 10.34 Let \( G \) be a graph with a 2-vertex cut \( \{ x, y\} \) . Then \( G \) is planar if and only if each of its marked \( \{ x, y\} \) -components is planar. | Proof Suppose, first, that \( G \) is planar. By Lemma 10.33, each marked \( \{ x, y\} \) - component of \( G \) is isomorphic to a minor of \( G \), hence is planar by Proposition 10.31.\n\nConversely, suppose that \( G \) has \( k \) marked \( \{ x, y\} \) -components each of which is planar. Let \( e \) denote their... | No |
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