# Unavoidable vertex-minors for large prime graphs O-joung Kwon, KAIST (This is a joint work with Sang-il Oum) Discrete Seminar in KAIST 1 October 4, 2013.

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Unavoidable vertex-minors for large prime graphs O-joung Kwon, KAIST (This is a joint work with Sang-il Oum) Discrete Seminar in KAIST 1 October 4, 2013. 1

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

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4 What if we add some connectivity assumtion?

5 Otherwise, the graph has bounded degree, so it must contain a long induced path. ■

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8 Over GF 2 !

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

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

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

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: Ladder : Generalized ladder => an induced cycle of length n ■ 26

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

33 1) K n 2) I n

34 1) K n 2) I n case is similar with K n case

35 1,2) It is equivalent to a path (w.r.t local complementations).

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37 Def. Linear rank-width of graphs is a complexity measure of graphs using the cut-rank function.

38 Prime notion is 2-rank connectivity. What will 3-rank connectivity say? Reduce the bound or find a simpler proof of our Theorem. ■

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