Unavoidable vertex-minors for large prime graphs

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

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

3. 1. Motivation 3

4. Ramsey type theorems Theorem (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 . What if we add some connectivityassumtion? Theorem (Diestel’s book) , , connected graphs on at least vertices => induced subgraphisomorphic to , or . 4

5. Ramsey type theorems Theorem (Diestel’s book) , , connected graphs on at least vertices => induced subgraphisomorphic to , or . 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. Therefore, we obtain or with the vertex Otherwise, the graph has bounded degree, so it must contain a long induced path. ■ 5

6. Theorem (Diestel’s book) , , 2 -connected graphs on at least vertices => a minor isomorphic to or. Theorem (Oporowski, Oxley, and Thomas 93) , , 3 -connected graphs on at least vertices => a minor isomorphic to or. 6

7. 2. What is cut-rank? 7

8. Def. Cut-rank of a vertex set ⊆ Over GF2 ! 8

9. Example 9

10. Cut-rank 1 Def. A vertex partition of is a split of G if the cut-rank of in is 1. Def. A prime graphis a graph having no split. (We sometimes call 2-rank connected) 10

11. Main Theorem Theorem (Oum,K) , every prime graph on at least vertices contains a vertex-minor isomorphic to ▤ or . ▤ 11

12. Vertex-minors Def. 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.

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

14. Outline of the proof 1. , , every prime graph having as a vertex-minor contains a vertex-minorisomorphic to . 2. , , every prime graph on vertices contains a vertex-minorisomorphic to or ▤ . 14

15. Known facts. 1. (fan) contains a vertex-minorisomorphic to . 15

16. 3. Blocking sequences (step 1) 16

17. Question. Whole graph is prime, but we have an induced subgraph which have a split. What can we say about the outside world? 17

18. Def. Def(Geelen’s Thesis 95). A sequence , … , iscalled a blocking sequence of a pair of disjoint subsets of if • , • for all , • , and • No proper subsequence of , … , satisfies (a), (b) and (c). 18

19. Example (Given graph is prime) 19

20. Example (In a prime graph) The sequence , also satisfies the conditions (a),(b),(c). Here, , is a blocking sequence.

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

22. Reduced blocking sequence 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 . 22

23. Example =>

24. Reduced blocking sequence 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. 24

25. 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 … 25

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

27. Outline of the proof 1. , , every prime graph having as a vertex-minor contains a vertex-minorisomorphic to . 2. , , every prime graph on vertices contains a vertex-minorisomorphic to or ▤ . 27

28. Known facts. 1. contains a vertex-minorisomorphic to. 2. contains a vertex-minorisomorphic to. Theorem (Diestel’s book) , , connected graphs on at least vertices => induced subgraphisomorphic to , or . Corollary , , connected graphs on at least vertices => vertex-minor isomorphic to. 28

29. 4. Brooms (step 2) 29

30. Def. -broom for . 30

31. Cor. , , every prime graph on vertices contains a vertex-minorisomorphic 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. 31

32. 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. 32

33. 1) Kn 2) In 33

34. 1) Kn 2) In case is similar with Kncase 34

35. 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. 35

36. With assuming that G has no or ▤vertex-minors, 1. , , -broom -> -broom 2. , , -broom -> -broom. 3. Finally, we obtains a vertex-minorisomorphic to . (arrive at -broom) huu… Bound is a super tower of exponential.. 36

37. 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 37

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

39. Thank you for listening! 39