# Structure of Graphs and Number of Maximum Matchings

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Structure of Graphs and Number of Maximum Matchings. Liu Yan School of Mathematics & South China Normal University. Topics. Gallai-Edmonds Decomposition Ear Decomposition Cathedral Construction. Gallai-Edmonds Decomposition. Basic Concepts: - PowerPoint PPT Presentation

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• Structure of Graphs and Number of Maximum MatchingsLiu Yan

School of Mathematics&South China Normal University

• TopicsGallai-Edmonds DecompositionEar DecompositionCathedral Construction

• Gallai-Edmonds DecompositionBasic Concepts:Factor-critical graph G: G-u has a p.m. for any u V(G).Bipartitie graph G=(X,Y) with positive surplus(about X): |(S)|>|S| for any SX.Deficiency of G: the number of vertices uncovered by a maximum matching, denoted by def(G).

• Gallai-Edmonds partition: Denote by D(G) the set of all vertices in G which are not covered by at least one maximum matching of G. Denote by A(G) the set of vertices in V(G)-D(G) adjacent to at least one vertex in D(G). C(G)=V(G)-A(G)-D(G).

• Gallai-Edmonds Decomposition Theorem

• For any maximum matching M of G, the number of non trivial components of which are M-near full is denoted by nc(M). Let nc(G)=nc= max{nc(M)} where the maximum is taken over all maximum matchings M.

Theorem (Liu Yan and Liu Guizhen, 2002 )For any graph G,f(G)=1/2(|V(G)|def(G)+nc) = 1/2(|V(G)|c(D(G))+|A(G)|+nc)

• Theorem (Liu Yan , submitted )Let B be a subset of E(G). If v(G-B)=v(G), then c(D(G-B))-|A(G-B)|=c(D(G))-|A(G)|; D(G-B) is a subset of D(G), C(G) is a subset of C(G-B)When G is bipartite, A(G-B) is a subset of A(G)

• Ear DecompositionLet G be a subgraph of a graph G. An ear: 1) a path P of odd length; 2) end-vertices of P are in V(G); 3) internal vertices of P are not in V(G).

An ear system: a set of vertex-disjoint ears.

Ear-decomposition: G = G1 G2 G3 Gr = Gwhere each Gi is an ear system and Gi+1 is obtained from Gi by an ear system so that Gi+1 is 1-extendable or factor-critical.

• Theorem (Liu Yan and Hao Jianxiu, 2002)Let G be a 2-connected factor-critical graph. Then G has precisely |E(G)| maximum matchings if and only if there exists an ear decomposition of G starting with a nice odd cycle C: G=C+P_1+ +P_k, satisfying that two ends of P_i are connected in G_{i-1} by a pending path of G with length 2, where G_0=C and G_i=C+P_1+ +P_i for 1i k.

• Theorem (Liu Yan , 2004 )

Let G be a factor-critical graph with c blocks. Then

(1) m(G)=|V(G)| if and only if every block of G is an odd cycle.

(2) m(G)=|V(G)|+1 if and only if all blocks of G are odd cycles but one, say such block H, and H is a theta graph (l1, l2, l3) with l1=2 and the path of length l1 is a pending path of G.

• Theorem (Liu Yan , 2006 )

Let G be a 2-connected factor-critical graph.Then m(G)=|V(G)|+2 if and only if G are the following.

• Cathedral ConstructionElementary graph: its allowed lines form a connected graph. Saturated graph G: G+e has more perfect matchings than G for all lines eE(G).Tuttes Theorem: A graph G has a p.m. if and only if o(G-S) |S| for any S V(G)Barrier set S: a vertex-set S satisfying o(G-S) = |S|

• Results about elementary graphsIf G is elementary, the set of maximal barriers is a partition of V(G).If G is saturated elementary and S is a maximal barrier, G has |S| maximal barriers which are singleton.If G is saturated elementary, the subgraph induced by a maximal barrier is complete.

• Cathedral construction of saturated graphs

• Theorem (Xiao Lan and Liu yan, 2006)1.Let G be an elementary graph with 2n vertices and having exactly two perfect matchings, then |E (G) |2n+n(n1)/2.2. If f(n) denotes the maximum number of edges in a graph on 2n vertices having exactly two perfect matchings, then f(n)=n2+1.

• Conjecture (Liu, 2006) If f(n) denotes the maximum number of edges in a graph on 2n vertices having exactly three perfect matchings, then f(n)=n2+2.