Decomposition Theory in Matching Covered Graphs

Qinglin YuQinglin Yu

Nankai U. , ChinaNankai U. , China

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U. C. Cariboo, CanadaU. C. Cariboo, Canada

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Topics

Basic conceptsBasic concepts Ear DecompositionEar Decomposition Brick DecompositionBrick Decomposition Matching LatticeMatching Lattice Properties of BricksProperties of Bricks

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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 SV(G)

Barrier set S: a vertex-set S satisfying o(G-S) = |S|

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1. Ear Decomposition

Let GLet G′′ be a subgraph of a graph G. be a subgraph of a graph G.

An An earear: 1) a path P of odd length; 2) end-vertices of P are in V(G: 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(3) internal vertices of P are not in V(GG′).′).

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

Ear-decomposition:Ear-decomposition:

GG′ =′ = G1 G2 G3 … … Gr = G= G

where each where each Gi is an ear system and is an ear system and Gi+1 is obtained from is obtained from Gi by an ear by an ear

system so that system so that Gi+1 is 1- is 1-extendableextendable..

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1. Ear Decomposition

Let GLet G′′ be a subgraph of a graph G. be a subgraph of a graph G.

An An earear: 1) a path P of odd length; 2) end-vertices of P are in V(G: 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(3) internal vertices of P are not in V(GG′).′).

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

Ear-decomposition:Ear-decomposition:

GG′ =′ = G1 G2 G3 … … Gr = G= G

where each where each Gi is an ear system and is an ear system and Gi+1 is obtained from is obtained from Gi by an ear by an ear

system so that system so that Gi+1 is 1- is 1-extendableextendable..

G’

P1 P2

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Theorem (Lovasz and Plummer, 76) Let G be a 1-extendable graph and GLet G be a 1-extendable graph and G′′ a subgraph of a subgraph of

G. Then G has an ear-decomposition starting G. Then G has an ear-decomposition starting with Gwith G′ if and only if G-V(′ if and only if G-V(GG′) has a p.m.′) has a p.m.

Theorem (Two Ears Theorem)

Every 1-extendable graph G has an ear Every 1-extendable graph G has an ear decomposition decomposition

K2 G2 G3 … … Gr = G= G

so that each so that each Gi contains at most two ears.contains at most two ears.

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Let d*(G) = min # of double ears in an ear Let d*(G) = min # of double ears in an ear decomposition of a graph G. decomposition of a graph G.

Optimal ear decompositionOptimal ear decomposition is an ear decomposition is an ear decomposition with exactly d*(G) double ears. with exactly d*(G) double ears.

Examples: i) a graph G is bipartite, then d*(G) = 0.Examples: i) a graph G is bipartite, then d*(G) = 0.

ii) For the Petersen graph P, d*(P) = 2. 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)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 where b(G) is # of bricks in G and p(G) is # of Petersen bricks in G. Petersen bricks in G. (Note: both b(G) and p(G) are invariants.)

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2. Brick DecompositionStep 1Step 1. . If G is a brick or is bipartite, then it is If G is a brick or is bipartite, then it is

indecomposableindecomposableStep 2Step 2. . Create bicriticality: if G is non-Create bicriticality: if G is non-

bipartite and not critical, then let X be a bipartite and not critical, then let X be a maximal barrier with |X| maximal barrier with |X| 2, and let S be 2, and let S be the vertex set of a component of G-X such the vertex set of a component of G-X such that |S| that |S| 3. Let 3. Let G1 = G = G S (the graph S (the graph obtained by shrinking S in G to a vertex) obtained by shrinking S in G to a vertex) and and G2 = G = G (G-S). Repeat this step on (G-S). Repeat this step on G1 and and G2..

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Brick Decomposition (Contin.)

Step 3Step 3. . Create 3-connectivity: if G is bicritical, but not Create 3-connectivity: if G is bicritical, but not 3-connected, then let {u, v} be a vertex cut, let S be 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 the vertex set of a component of G-{u, v} and let T = V-(S= V-(S{u, v}). Let G{u, v}). Let G1 = G = G (S (S {u}) and G{u}) and G2 = =

G G (T (T {v}). Repeat this step on G{v}). Repeat this step on G11 and G and G22..

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Example: Example:

Step 2:Step 2:

Step 3:Step 3:

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Theorem (Lovasz, 87) In a brick decomposition, the list of bricks and In a brick decomposition, the list of bricks and

bipartite graphs are independent or unique (up to bipartite graphs are independent or unique (up to multiplicity of edges) of choices of max. barrier multiplicity of edges) of choices of max. barrier X (in Step 2) or 2-cut (in Step 3).X (in Step 2) or 2-cut (in Step 3).

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3. Matching LatticeFor any A For any A E(G), incident vector E(G), incident vector χχAA of A in of A in ZZEE is a vector is a vector

w of 0’s and 1’s such that w(e) = 0 if ew of 0’s and 1’s such that w(e) = 0 if eA and w(e) = 1 A and w(e) = 1 if e if e A. A.

Matching latticeMatching lattice for G=(V, E) is denoted for G=(V, E) is denoted

Lat(G) := {wLat(G) := {w ZZEE : w = : w = MMMM M M χχMM, , M M ZZ }}

IntCon(G) := {wIntCon(G) := {w ZZEE00 : w = : w = MMMM M M χχ

MM, , M M ZZ0 0 }}

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Matching Lattice (Contin.)

Example:Example:

H=CH=C33KK22 is a 1-extendable and is a 1-extendable and

bicritical graph (i.e., a brick)bicritical graph (i.e., a brick) It has only 4 p.m., namely It has only 4 p.m., namely

MM11={1, 5, 9}, ={1, 5, 9}, χχMM11 = {1,0,0,0,1,0,0,0,1}; = {1,0,0,0,1,0,0,0,1};

MM22={2, 6, 7}, ={2, 6, 7}, χχMM22 = {0,1,0,0,0,1,1,0,0}; = {0,1,0,0,0,1,1,0,0};

MM33={3, 4, 8}, ={3, 4, 8}, χχMM33 = {0,0,1,1,0,0,0,1,0}; = {0,0,1,1,0,0,0,1,0};

MM44={4, 5, 6}, ={4, 5, 6}, χχMM44 = {0,0,0,1,1,1,0,0,0}; = {0,0,0,1,1,1,0,0,0};

Lat(H) is all integer combinations of Lat(H) is all integer combinations of χχMM11, , χχMM22, , χχMM33 and and χχMM44

11

9

87

5

32

64

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Remarks: Remarks: Convex hull, integer cone and lattice have been Convex hull, integer cone and lattice have been used as tools or relaxations to many well known problemsused as tools or relaxations to many well known problems

1)1) A k-regular graph G is k-edge-colorable A k-regular graph G is k-edge-colorable 1-factorable 1-factorable 11= {1, 1, …, 1} lies in IntCon(G); = {1, 1, …, 1} lies in IntCon(G); 2)2) 4CC 4CC every 2-connected cubic planar graph is 3-edge every 2-connected cubic planar graph is 3-edge

colorable colorable 11 IntCon; IntCon;3)3) For an ear decompositionFor an ear decomposition

GG = K = K2 G2 G3 … … Gr = G = G

In matching lattice, we can associate each subgraph GIn matching lattice, we can associate each subgraph Gii with a p.m., to obtain a set of r p.m.with a p.m., to obtain a set of r p.m.′′s Ms M11, M, M22, …, M, …, Mrr so so

that that χχMM11, , χχMM22, …, , …, χχMMrr are linearly independent. are linearly independent.

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Theorem (Edmonds, Lovasz and Pulleyblank, 82)

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

Theorem (Lovasz, 87)

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

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Theorem (Lovasz, 87)

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

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4. Structure results of bricksTheorem (Carvahho, et al, '02)

Every brick G ( KK44, CC33KK22) has (D-2) edges eii’s such that G-eii is matching-covered.

Conjecture (Lovasz, 87)Every brick different from KK44, CC33KK22, 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.)

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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{|MC|: Mm(G)} = 3 or 5.

Theorem

If min{|MC|: Mm(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)

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Theorem Let G be a brick and its ear-decomposition is

G1 G2 G3 … … Gr-1 Gr = G = GThen either Gr-1 is bipartite or G arises from Gr-1 by adding a single edge.

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