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Unavoidable vertex-minors for

large prime graphsO-joung Kwon, KAIST

(This is a joint work with Sang-il Oum)

Discrete Seminar in KAIST

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October 4, 2013.

Index- Ramsey type theorems - Rank-connectivity- Main Theorem 1) Blocking sequences Ladders 2) Ramsey type argument Brooms- What is next?

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1. Motivation

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Ramsey type theoremsTheorem (Ramsey theorem) For every , there exists such that every graph on at least vertices contains an induced subgraph isomorphic to

or the graph consists of the independent set of size .

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What if we add some connectivity assumtion?Theorem (Diestel’s book), , connected graphs on at least vertices => induced subgraph isomorphic to , or .

Ramsey type theorems

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Proof) If a vertex in a graph has degree at least

by Ramsey Theorem, we have or an independent set of size in the

neighbors.

Theorem (Diestel’s book), , connected graphs on at least vertices => induced subgraph isomorphic to , or .

Therefore, we obtain or with the vertex Otherwise, the graph has bounded degree,

so it must contain a long induced path. ■

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Theorem (Oporowski, Oxley, and Thomas 93), , 3 -connected graphs on at least vertices => a minor isomorphic to or.

Theorem (Diestel’s book), , 2 -connected graphs on at least vertices => a minor isomorphic to or.

2. What is cut-rank?

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Def. Cut-rank of a vertex set ⊆

Over GF2 !

Example

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Cut-rank 1

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Def. A prime graph is a graph having no split. (We sometimes call 2-rank connected)

Def. A vertex partition of is a split of Gif the cut-rank of in is 1.

Main Theorem

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Theorem (Oum,K) , every prime graph on at least vertices contains a vertex-minor isomorphic to ▤ or .

▤

Vertex-minorsDef. A local complementation at a vertex in () is an operation to flip the edges between the neighbors of Def. A graph is a vertex-minor of if can be obtained from by applying local complementations and vertex deletions.

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Why local complementation?

* Local complementation preserves the cut-rank of all vertex subsets.

Outline of the proof

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1. , , every prime graph having as a vertex-minor contains a vertex-minor isomorphic to .

2. , , every prime graph on vertices contains a vertex-minor isomorphic to or ▤ .

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Known facts.1. (fan) contains a vertex-minor isomorphic to .

3. Blocking sequences (step 1)

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Question. Whole graph is prime, but we have an induced subgraph which

have a split. What can we say about the outside world?

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Def(Geelen’s Thesis 95). A sequence , … , is called a blocking sequence of a pair of disjoint subsets of if(a) , (b) for all ,(c) , and(d) No proper subsequence of , … , satisfies (a), (b)

and (c).

Def.

Example (Given graph is prime)

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Example (In a prime graph)

The sequence , also satisfies the conditions (a),(b),(c).

Here, , is a blocking sequence.

Assuming existence of a long induced path. Object : obtaining a sufficiently long induced cycle.

…

…

Reduced blocking sequence

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Proposition. Let , … , be a blocking sequence for in . Let

1) If then a sequence , … , is a blocking sequence for in for each .

2) If then .

Example

=>

Reduced blocking sequence

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Proposition. : prime, : a cut of cut-rank 1 in an induced subgraph.

Then there exists a graph equivalent to (w.r.t local. c.)

such that

1) ,2) has a blocking sequence of length at most 6.

Each step) 1) Take a reduced blocking sequence 2) Reduce this blocking sequence to one vertex by spending some vertices on the path (without destroying “the path” / # is bounded ! ) 3) Make adjacent to only one vertex in the right cut

…

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: Ladder

: Generalized ladder

=> an induced cycle of length n ■ 26

Outline of the proof

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1. , , every prime graph having as a vertex-minor contains a vertex-minor isomorphic to .

2. , , every prime graph on vertices contains a vertex-minor isomorphic to or ▤ .

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Corollary , , connected graphs on at least vertices => vertex-minor isomorphic to.

Known facts.1. contains a vertex-minor isomorphic to.

2. contains a vertex-minor isomorphic to.Theorem (Diestel’s book), , connected graphs on at least vertices => induced subgraph isomorphic to , or .

4. Brooms (step 2)

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Def. -broom for .

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Cor. , , every prime graph on vertices contains a vertex-minor isomorphic to -broom.Observ. Any two leaves of -broom in a prime gr,

have different kind of neighbors in the outside,

otherwise they form a split.

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Observ. , such that a prime graph having -broom contains one of the following of width .

By Ramsey Theorem, we may assume that second vertices form a clique or an independent set.

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1) Kn

2) In

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1) Kn

2) In case is similar with Kn

case

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1,2) It is equivalent to a path (w.r.t local complementations).

Observ) Only remaining graph is -broom.

Prop. , , every prime graph having -broom contains a vertex-minor isomorphic to

either or ▤ or -broom.

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With assuming that G has no or ▤ vertex-minors,

3. Finally, we obtains a vertex-minor isomorphic to . (arrive at -broom) huu… Bound is a super tower of exponential..

1. , , -broom -> -broom

2. , , -broom -> -broom.

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Def. Linear rank-width of graphs is a complexity measure of graphs using the cut-rank function.Vertex ordering (, … , ) of Width of an ordering : maximum of all , … , Linear rank-width : minimum width of all possible orderings.Conjecture (Courcelle; Oum)For any fixed tree , there exists a function s.t,if a graph has linear rank width at least , it contains a vertex-minor isomorphic to .We don’t know yet even for paths instead of trees

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Prime notion is 2-rank connectivity.What will 3-rank connectivity say?

Reduce the bound or find a simpler proof of our Theorem. ■

I believe one would be

Thank you for listening!

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