Decomposition Theory in Matching Covered Graphs

Decomposition Theory in Matching Covered Graphs Qinglin Yu Nankai U. , China & U. C. Cariboo, Canada

Topics • Basic concepts • Ear Decomposition • Brick Decomposition • Matching Lattice • Properties of Bricks

Basic Concepts: Perfect Matching (p.m.): a set of independent edges saturated all vertices of a graph. 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). 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|

1. 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′ =G1G2G3 … Gr= G where each Gi is an ear system and Gi+1is obtained from Gi by an ear system so that Gi+1 is 1-extendable.

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 K2G2G3 … Gr= G so that each Gicontains at most two ears.

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. (Note: both b(G) and p(G) are invariants.)

2. 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 G1 = G  S (the graph obtained by shrinking S in G to a vertex) and G2 = G  (G-S). Repeat this step on G1 and G2.

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 G1 = G  (S {u}) and G2 = G  (T {v}). Repeat this step on G1 and G2.

Example: Step 2: Step 3:

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).

3. Matching Lattice For any A  E(G), incident vector χA of A in ZE is a vector w of 0’s and 1’s such that w(e) = 0 if eA and w(e) = 1 if e A. Matching lattice for G=(V, E) is denoted Lat(G) := {w ZE : w = MMM χM, M Z} IntCon(G) := {w ZE0 : w = MMM χM, M Z0 }

Matching Lattice (Contin.) • Example: H=C3K2 is a 1-extendable and bicritical graph (i.e., a brick) • It has only 4 p.m., namely M1={1, 5, 9}, χM1 = {1,0,0,0,1,0,0,0,1}; M2={2, 6, 7}, χM2 = {0,1,0,0,0,1,1,0,0}; M3={3, 4, 8}, χM3 = {0,0,1,1,0,0,0,1,0}; M4={4, 5, 6}, χM4 = {0,0,0,1,1,1,0,0,0}; Lat(H) is all integer combinations of χM1, χM2, χM3and χM4

Remarks: Convex hull, integer cone and lattice have been used as tools or relaxations to many well known problems • A k-regular graph G is k-edge-colorable  1-factorable 1= {1, 1, …, 1} lies in IntCon(G); • 4CC  every 2-connected cubic planar graph is 3-edge colorable 1 IntCon; • For an ear decomposition G = K2G2G3 … Gr= G In matching lattice, we can associate each subgraph Gi with a p.m., to obtain a set of r p.m.′s M1, M2, …, Mr so that χM1, χM2, …, χMrare linearly independent.

Theorem (Edmonds, Lovasz and Pulleyblank, 82) Let G = (V, E) be a 1-extendable graph. Let P(G) be p.m. polytope (i.e., convex hull of incident vectors of p.m. of G). Then the dimension of P(G) is |E|-|V|-1 if G is bipartite; |E|- |V| +1 – b if G is nonbipartite (where b is # of bricks in brick decomposition) Theorem (Lovasz, 87) Let G = (V, E) be a 1-extendable graph. Then the dimension of Lat(G) is |E|-|V|+2 – b.

Theorem (Lovasz, 87) Let G = (V, E) be a 1-extendable graph. Then the dimension of Lat(G) is |E|-|V|+2 – b.

4. Structure results of bricks Theorem (Carvahho, et al, '02) Every brick G ( K4, C3K2) has (D-2) edges ei’s such that G-ei is matching-covered. Conjecture (Lovasz, 87) Every brick different from K4, C3K2, and Petersen graph has an edge e such that G and G-e have the same number of bricks. (Carvahho proved this conjecture and showed a strong result that there exists an edge e such that (b+p)(G-e) = (b+p)(G), where b is # of bricks and p is # of Petersen bricks.)

Theorem Let G be a 1-extendable graph and C is a vertex-cut of G so that both C-contractions are 1-extendable. If C is not tight, then min{|MC|: Mm(G)} = 3 or 5. Theorem If min{|MC|: Mm(G)} = 5, then the underlying simple graph of G is Petersen graph. (Lovasz’s conjecture that every minimal (edge-wise) brick has two adjacent vertices of degree 3 is still open)

Theorem Let G be a brick and its ear-decomposition is G1G2G3 … Gr-1Gr= G Then either Gr-1 is bipartite or G arises from Gr-1 by adding a single edge.

Thank you!!

Decomposition Theory in Matching Covered Graphs