Matching Theory and Structure of Graphs Liu Yan School of Mathematics & South China Normal University

7/20/20042 Topics  Basic concepts  Gailei-Edmonds Decomposition  Ear Decomposition  Brick Decomposition

7/20/20043 Basic Concepts: Matching-covered (or 1-extendable) graph: every edge lies in a p.m. Bicritical graph G: G-{u, v} has a p.m. for any pair of {u, v}  V(G). Factor-critical graph G: G-u has a p.m. for any u V(G). Tutte’s 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|

7/20/ 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). Let C(G)=V(G)- A(G)-D(G). We call the partition (C(G),A(G),D(G)) the Gallai-Edmonds partition

7/20/20045  Theorem ( Liu Yan and Liu Guizhen, 2002 )  For any graph G,  V_f(G)=1/2(|V(G)|-max{i(G-S)-|S|})  where the maximum is taken over all subsets S of V(G).  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)

7/20/ Ear Decomposition Let 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′ = G 1  G 2  G 3  …  G r = G where each G i is an ear system and G i+1 is obtained from G i by an ear system so that G i+1 is 1-extendable or factor-critical.

7/20/20047 Theorem ( Lovasz and Plummer, 76 ) Let G be a 1-extendable graph and G′ a subgraph of G. Then G has an ear-decomposition starting with G′ if and only if G-V(G′) has a p.m. Theorem ( Two Ears Theorem ) Every 1-extendable graph G has an ear decomposition    …  = G K 2  G 2  G 3  …  G r = G so that each contains at most two ears. so that each G i contains at most two ears.

7/20/20048 Let d*(G) = min # of double ears in an ear decomposition of a graph G. Optimal ear decomposition is an ear decomposition with exactly d*(G) double ears. Examples: i) a graph G is bipartite, then d*(G) = 0. ii) For the Petersen graph P, d*(P) = 2. Theorem ( Carvahho, et al, '02 ) If G is a 1-extenable graph, then d*(G) = b(G) + p(G) where b(G) is # of bricks in G and p(G) is # of Petersen bricks in G. where b(G) is # of bricks in G and p(G) is # of Petersen bricks in G. (Note: both b(G) and p(G) are invariants.)

7/20/20049 Theorem ( Liu Yan and Hao Jianxiu, 2002 ) Let G be a factor-critical graph and c the number of blocks of G. Then G has precisely |E(G)|-c+1 near-perfect matchings if and only if there exists an ear decomposition of G starting with a nice odd cycle C; that is G=C+P_1+…… +P_k, satisfying that two ends of P_iare connected in G_{i-1} by a pending path of G with length 2 if P_i is open, where G_0=C and G_i=C+P_1+\cdots +P_i for 1≤i≤ k.

7/20/ Theorem ( Liu Yan, submitted ) 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_1, and H_1 is a theta graph θ(l_1, l_2, l_3) with common endvertices u and v satisfying that l_1=2 and the path of length l_1 in H_1 joining u and v is a pending path of G.

7/20/ Brick Decomposition Step 1. If G is a brick or is bipartite, then it is indecomposable Step 2. Create bicriticality: if G is non-bipartite and not critical, then let X be a maximal barrier with |X|  2, and let S be the vertex set of a component of G-X such that |S|  3. Let G 1 = G  S (the graph obtained by shrinking S in G to a vertex) and G 2 = G  (G-S). Repeat this step on G 1 and G 2.

7/20/ Brick Decomposition (Contin.) Step 3. Create 3-connectivity: if G is bicritical, but not 3-connected, then let {u, v} be a vertex cut, let S be the vertex set of a component of G-{u, v} and let T = V-(S  {u, v}). Let G 1 = G  (S  {u}) and G 2 = G  (T  {v}). Repeat this step on G 1 and G 2.

7/20/ Example: Step 2: Step 3:

7/20/ Theorem (Lovasz, 87) In a brick decomposition, the list of bricks and bipartite graphs are independent or unique (up to multiplicity of edges) of choices of max. barrier X (in Step 2) or 2-cut (in Step 3).

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