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

7/20/20042 Topics Basic concepts Basic concepts Ear Decomposition Ear Decomposition Brick Decomposition Brick Decomposition Matching Lattice Matching Lattice Properties of Bricks Properties of Bricks

7/20/20043 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|

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. An ear system: a set of vertex-disjoint ears. Ear-decomposition: Ear-decomposition: G′ =    …  = G G′ = G 1  G 2  G 3  …  G r = G where each is an ear system and is obtained from by an ear system so that is 1-extendable. 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.

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. An ear system: a set of vertex-disjoint ears. Ear-decomposition: Ear-decomposition: G′ =    …  = G G′ = G 1  G 2  G 3  …  G r = G where each is an ear system and is obtained from by an ear system so that is 1-extendable. 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.

7/20/20046 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.

7/20/20047 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. 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/ 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  S (the graph obtained by shrinking S in G to a vertex) and = G  (G-S). Repeat this step on and. 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.

7/20/20049 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 = G  (S  {u}) and G = G  (T  {v}). Repeat this step on G 1 and G 2. 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).

7/20/ Matching Lattice For any A  E(G), incident vector χ A of A in Z E 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  Z E : w =  M  M  M χ M,  M  Z } IntCon(G) := {w  Z E  0 : w =  M  M  M χ M,  M  Z  0 }

7/20/ Matching Lattice (Contin.) Example: Example: H=C 3  K 2 is a 1-extendable and H=C 3  K 2 is a 1-extendable and bicritical graph (i.e., a brick) It has only 4 p.m., namely It has only 4 p.m., namely M 1 ={1, 5, 9}, χ M 1 = {1,0,0,0,1,0,0,0,1}; M 2 ={2, 6, 7}, χ M 2 = {0,1,0,0,0,1,1,0,0}; M 3 ={3, 4, 8}, χ M 3 = {0,0,1,1,0,0,0,1,0}; M 4 ={4, 5, 6}, χ M 4 = {0,0,0,1,1,1,0,0,0}; Lat(H) is all integer combinations of χ M 1, χ M 2, χ M 3 and χ M 4 Lat(H) is all integer combinations of χ M 1, χ M 2, χ M 3 and χ M 4

7/20/ Remarks: Convex hull, integer cone and lattice have been used as tools or relaxations to many well known problems 1) A k-regular graph G is k-edge-colorable  1-factorable  1= {1, 1, …, 1} lies in IntCon(G); 1-factorable  1= {1, 1, …, 1} lies in IntCon(G); 2) 4CC  every 2-connected cubic planar graph is 3-edge colorable  1  IntCon; 3) For an ear decomposition G = K    …  = G G = K 2  G 2  G 3  …  G r = G In matching lattice, we can associate each subgraph G i with a p.m., to obtain a set of r p.m.′s M 1, M 2, …, M r so that χ M 1, χ M 2, …, χ M r are linearly independent.

7/20/ 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.

7/20/ Theorem (Lovasz, 87) Let G = (V, E) be a 1-extendable graph. Then the dimension of Lat(G) is |E|-|V|+2 – b.

7/20/ Structure results of bricks Theorem (Carvahho, et al, '02) K 4 C 3  K 2i i Every brick G (  K 4, C 3  K 2 ) has (D-2) edges e i ’s such that G-e i is matching-covered. Conjecture (Lovasz, 87) K 4 C 3  K 2 Every brick different from K 4, C 3  K 2, 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.)

7/20/ 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)

7/20/ Theorem Let G be a brick and its ear-decomposition is    …  = G G 1  G 2  G 3  … G r-1  G r = G Then either G r-1 is bipartite or G arises from G r-1 by adding a single edge.

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