K 4 -based and C 6 -based planar bricks Nishad Kothari - - PowerPoint PPT Presentation

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K 4 -based and C 6 -based planar bricks Nishad Kothari - - PowerPoint PPT Presentation

Apr 03, 2023 358 likes •962 views

Motivation Our Results K 4 -based and C 6 -based planar bricks Nishad Kothari nkothari@math.uwaterloo.ca (joint work with U. S. R. Murty) June 10, 2013 @ CanaDAM K 4 C 6 K 4 -based and C 6 -based planar bricks Motivation Our Results Matching

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Motivation Our Results

K4-based and C6-based planar bricks

Nishad Kothari

nkothari@math.uwaterloo.ca

(joint work with U. S. R. Murty)

June 10, 2013 @ CanaDAM K4 C6

K4-based and C6-based planar bricks

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Motivation Our Results

Matching Covered Graphs

K4-based and C6-based planar bricks

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Motivation Our Results

Matching Covered Graphs

Petersen

K4-based and C6-based planar bricks

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Motivation Our Results

Matching Covered Graphs

Petersen Definition (Matching Covered Graph) A connected graph with at least two vertices is matching covered if each of its edges lies in some perfect matching.

K4-based and C6-based planar bricks

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Motivation Our Results

Matching Covered Graphs

Petersen Definition (Matching Covered Graph) A connected graph with at least two vertices is matching covered if each of its edges lies in some perfect matching. For example, any 2-connected cubic graph is matching covered.

K4-based and C6-based planar bricks

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Motivation Our Results

A classification of nonbipartite matching covered graphs

K4-based and C6-based planar bricks

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Motivation Our Results

A classification of nonbipartite matching covered graphs

K4-based C6-based

K4-based and C6-based planar bricks

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Motivation Our Results

Building a matching covered graph

K4-based and C6-based planar bricks

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Motivation Our Results

Building a matching covered graph

G

K4-based and C6-based planar bricks

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Motivation Our Results

Building a matching covered graph

G G1

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Motivation Our Results

Building a matching covered graph

G

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Motivation Our Results

Building a matching covered graph

G

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Motivation Our Results

Building a matching covered graph

G

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Motivation Our Results

Building a matching covered graph

G G2

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Motivation Our Results

Building a matching covered graph

G G2

K4-based and C6-based planar bricks

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Motivation Our Results

Ear decomposition

Definition (Single and double ears) A single ear of a graph is a path of odd length whose each internal vertex (if any) has degree two in the graph.

K4-based and C6-based planar bricks

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Motivation Our Results

Ear decomposition

Definition (Single and double ears) A single ear of a graph is a path of odd length whose each internal vertex (if any) has degree two in the graph. A pair of vertex-disjoint single ears is called a double ear.

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Motivation Our Results

Ear decomposition

Definition (Single and double ears) A single ear of a graph is a path of odd length whose each internal vertex (if any) has degree two in the graph. A pair of vertex-disjoint single ears is called a double ear.

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Motivation Our Results

Ear decomposition

Definition (Single and double ears) A single ear of a graph is a path of odd length whose each internal vertex (if any) has degree two in the graph. A pair of vertex-disjoint single ears is called a double ear.

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Motivation Our Results

Ear decomposition

Definition (Single and double ears) A single ear of a graph is a path of odd length whose each internal vertex (if any) has degree two in the graph. A pair of vertex-disjoint single ears is called a double ear.

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Motivation Our Results

Ear decomposition

Definition (Single and double ears) A single ear of a graph is a path of odd length whose each internal vertex (if any) has degree two in the graph. A pair of vertex-disjoint single ears is called a double ear.

K4-based and C6-based planar bricks

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Motivation Our Results

Ear decomposition

Definition (Ear decomposition) An ear decomposition of a matching covered graph G is a sequence G1 ⊂ G2 ⊂ ... ⊂ Gr of matching covered subgraphs of G such that:

K4-based and C6-based planar bricks

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Motivation Our Results

Ear decomposition

Definition (Ear decomposition) An ear decomposition of a matching covered graph G is a sequence G1 ⊂ G2 ⊂ ... ⊂ Gr of matching covered subgraphs of G such that:

1 for each i such that 1 ≤ i ≤ r − 1, Gi+1 is the union of Gi

and exactly one single ear or double ear of Gi+1, and

K4-based and C6-based planar bricks

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Motivation Our Results

Ear decomposition

Definition (Ear decomposition) An ear decomposition of a matching covered graph G is a sequence G1 ⊂ G2 ⊂ ... ⊂ Gr of matching covered subgraphs of G such that:

1 for each i such that 1 ≤ i ≤ r − 1, Gi+1 is the union of Gi

and exactly one single ear or double ear of Gi+1, and

2 Gr := G.

K4-based and C6-based planar bricks

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Motivation Our Results

Ear decomposition theorem

Theorem (Lovász) Every nonbipartite matching covered graph G has an ear decomposition G1 ⊂ G2 ⊂ ... ⊂ Gr such that G1 is either:

K4-based and C6-based planar bricks

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Motivation Our Results

Ear decomposition theorem

Theorem (Lovász) Every nonbipartite matching covered graph G has an ear decomposition G1 ⊂ G2 ⊂ ... ⊂ Gr such that G1 is either:

1 a bi-subdivision of K4

K4-based and C6-based planar bricks

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Motivation Our Results

Ear decomposition theorem

Theorem (Lovász) Every nonbipartite matching covered graph G has an ear decomposition G1 ⊂ G2 ⊂ ... ⊂ Gr such that G1 is either:

1 a bi-subdivision of K4 (we say G is K4-based), or

K4-based and C6-based planar bricks

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Motivation Our Results

Ear decomposition theorem

Theorem (Lovász) Every nonbipartite matching covered graph G has an ear decomposition G1 ⊂ G2 ⊂ ... ⊂ Gr such that G1 is either:

1 a bi-subdivision of K4 (we say G is K4-based), or 2 a bi-subdivision of C6

K4-based and C6-based planar bricks

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Motivation Our Results

Ear decomposition theorem

Theorem (Lovász) Every nonbipartite matching covered graph G has an ear decomposition G1 ⊂ G2 ⊂ ... ⊂ Gr such that G1 is either:

1 a bi-subdivision of K4 (we say G is K4-based), or 2 a bi-subdivision of C6 (we say G is C6-based).

K4-based and C6-based planar bricks

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Motivation Our Results

K4-based C6-based

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Motivation Our Results

K4-based C6-based

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Motivation Our Results

K4-based C6-based

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Motivation Our Results

K4-based C6-based

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Motivation Our Results

K4-based C6-based

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Motivation Our Results

K4-based C6-based

K6

K4-based and C6-based planar bricks

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Motivation Our Results

Bricks

Definition (Brick) A 3-connected bicritical graph is called a brick. (These are special nonbipartite matching covered graphs.)

K4-based and C6-based planar bricks

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Motivation Our Results

Bricks

Definition (Brick) A 3-connected bicritical graph is called a brick. (These are special nonbipartite matching covered graphs.)

K4-based and C6-based planar bricks

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Motivation Our Results

Bricks

Definition (Brick) A 3-connected bicritical graph is called a brick. (These are special nonbipartite matching covered graphs.) Corollary Every brick is either K4-based or C6-based (or possibly both).

K4-based and C6-based planar bricks

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Motivation Our Results

A necessary condition for a planar brick to be K4-based

Let G be a plane matching covered graph which is K4-based.

K4-based and C6-based planar bricks

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Motivation Our Results

A necessary condition for a planar brick to be K4-based

Let G be a plane matching covered graph which is K4-based. Then G has an ear decomposition G1 ⊂ G2 ⊂ ... ⊂ Gr such that G1 is a bi-subdivision of K4.

K4-based and C6-based planar bricks

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Motivation Our Results

A necessary condition for a planar brick to be K4-based

Let G be a plane matching covered graph which is K4-based. Then G has an ear decomposition G1 ⊂ G2 ⊂ ... ⊂ Gr such that G1 is a bi-subdivision of K4. Note that K4 has exactly four odd faces, and so does G1.

K4-based and C6-based planar bricks

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Motivation Our Results

A necessary condition for a planar brick to be K4-based

Let G be a plane matching covered graph which is K4-based. Then G has an ear decomposition G1 ⊂ G2 ⊂ ... ⊂ Gr such that G1 is a bi-subdivision of K4. Note that K4 has exactly four odd faces, and so does G1. We conclude that G has at least four odd faces.

K4-based and C6-based planar bricks

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Motivation Our Results

A necessary condition for a planar brick to be K4-based

Let G be a plane matching covered graph which is K4-based. Then G has an ear decomposition G1 ⊂ G2 ⊂ ... ⊂ Gr such that G1 is a bi-subdivision of K4. Note that K4 has exactly four odd faces, and so does G1. We conclude that G has at least four odd faces.

K4-based and C6-based planar bricks

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Motivation Our Results

A necessary condition for a planar brick to be K4-based

Let G be a plane matching covered graph which is K4-based. Then G has an ear decomposition G1 ⊂ G2 ⊂ ... ⊂ Gr such that G1 is a bi-subdivision of K4. Note that K4 has exactly four odd faces, and so does G1. We conclude that G has at least four odd faces. Proposition A K4-based planar brick must have at least four odd faces.

K4-based and C6-based planar bricks

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Motivation Our Results

K4-based planar bricks

K4-based and C6-based planar bricks

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Motivation Our Results

K4-based planar bricks

Theorem (K. and Murty) Let G be a planar brick. If G has at least four odd faces then G is K4-based.

K4-based and C6-based planar bricks

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Motivation Our Results

K4-based planar bricks

Theorem (K. and Murty) Let G be a planar brick. If G has at least four odd faces then G is K4-based. In other words, the K4-free planar bricks are precisely those which have exactly two odd faces.

K4-based and C6-based planar bricks

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Motivation Our Results

Two infinite families of C6-free planar bricks

K4-based and C6-based planar bricks

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Motivation Our Results

Two infinite families of C6-free planar bricks

An odd wheel

K4-based and C6-based planar bricks

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Motivation Our Results

Two infinite families of C6-free planar bricks

An odd wheel An odd staircase

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Motivation Our Results

Two infinite families of C6-free planar bricks

An odd wheel An odd staircase

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Motivation Our Results

Two infinite families of C6-free planar bricks

An odd wheel An odd staircase

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Motivation Our Results

Two infinite families of C6-free planar bricks

An odd wheel An odd staircase

K4-based and C6-based planar bricks

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Motivation Our Results

C6-based planar bricks

Tricorn

K4-based and C6-based planar bricks

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Motivation Our Results

C6-based planar bricks

Tricorn Theorem (K. and Murty) Let G be a planar brick. If G is not an odd wheel or an odd staircase or the Tricorn, then G is C6-based.

K4-based and C6-based planar bricks

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Motivation Our Results

C6-based planar bricks

Tricorn Theorem (K. and Murty) Let G be a planar brick. If G is not an odd wheel or an odd staircase or the Tricorn, then G is C6-based. In other words, the C6-free planar bricks are precisely the odd wheels, the odd staircases and the Tricorn.

K4-based and C6-based planar bricks

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Motivation Our Results

Why tricorn?

Tricorn

K4-based and C6-based planar bricks

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Motivation Our Results

Why tricorn?

Tricorn Tricorn

K4-based and C6-based planar bricks