Decomposition Theory in
Matching Covered Graphs
Qinglin Yu
Nankai U. , China
&
U. C. Cariboo, Canada
Topics
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Basic concepts
Ear Decomposition
Brick Decomposition
Matching Lattice
Properties 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 S V(G)
Barrier set S: a vertex-set S satisfying o(G-S) = |S|
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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+1 is obtained from Gi by an ear
system so that Gi+1 is 1-extendable.
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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’
P1
P2
G′ = G1  G2  G3  …  Gr = G
where each Gi is an ear system and Gi+1 is obtained from Gi by an ear
system so that Gi+1 is 1-extendable.
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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 Gi contains at most two ears.
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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.)
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2. Brick Decomposition
Step 1. If G is a brick or is bipartite, then it is
indecomposable
Step 2. Create bicriticality: if G is nonbipartite 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.
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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.
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Example:
Step 2:
Step 3:
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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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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 }
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1
Matching Lattice (Contin.)
3
2
Example:
6
4
5
H=C3K2 is a 1-extendable and
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bicritical graph (i.e., a brick)
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 It has only 4 p.m., namely
M1={1, 5, 9}, χM = {1,0,0,0,1,0,0,0,1};
M2={2, 6, 7}, χM = {0,1,0,0,0,1,1,0,0};
M3={3, 4, 8}, χM = {0,0,1,1,0,0,0,1,0};
M4={4, 5, 6}, χM = {0,0,0,1,1,1,0,0,0};
Lat(H) is all integer combinations of χM , χM , χM and

1
2
3
4
1
2
3
χM
4
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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);
2)
4CC  every 2-connected cubic planar graph is 3-edge
colorable  1 IntCon;
3)
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 χM , χM , …, χM are linearly independent.
1
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r
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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.
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Theorem (Lovasz, 87)
Let G = (V, E) be a 1-extendable graph. Then the
dimension of Lat(G) is |E|-|V|+2 – b.
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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.)
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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,
m
then min{|MC|: M (G)} = 3 or 5.
Theorem
m
If min{|MC|: 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)
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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.
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