1. 1 Almost all cop-win graphs contain a universal vertex Anthony Bonato Ryerson University CanaDAM 2011

2. Cop number of a graph the cop number of a graph, written c(G), is an elusive graph parameter –few connections to other graph parameters –hard to compute –hard to find bounds –structure of k-cop-win graphs with k > 1 is not well understood Random cop-win graphs Anthony Bonato 2

3. Cops and Robbers played on reflexive graphs G two players Cops C and robber R play at alternate time-steps (cops first) with perfect information players move to vertices along edges; allowed to moved to neighbors or pass cops try to capture (i.e. land on) the robber, while robber tries to evade capture minimum number of cops needed to capture the robber is the cop number c(G) –well-defined as c(G) ≤ γ(G) Random cop-win graphs Anthony Bonato 3

4. Fast facts about cop number (Aigner, Fromme, 84) introduced parameter –G planar, then c(G) ≤ 3 (Berrarducci, Intrigila, 93), (Hahn, MacGillivray,06), (B, Chiniforooshan,10): “c(G) ≤ s?” s fixed: running time O(n 2s+3 ), n = |V(G)| (Fomin, Golovach, Kratochvíl, Nisse, Suchan, 08): if s not fixed, then computing the cop number is NP-hard (Shroeder,01) G genus g, then c(G) ≤ ⌊ 3g/2 ⌋ +3 (Joret, Kamiński, Theis, 09) c(G) ≤ tw(G)/2 Random cop-win graphs Anthony Bonato 4

5. Meyniel’s Conjecture c(n) = maximum cop number of a connected graph of order n Meyniel Conjecture: c(n) = O(n 1/2 ). Random cop-win graphs Anthony Bonato 5

6. State-of-the-art (Lu, Peng, 09+) proved that independently proved by (Scott, Sudakov,10+), and (Frieze, Krivelevich, Loh, 10+) even proving c(n) = O(n 1-ε ) for some ε > 0 is open Random cop-win graphs Anthony Bonato 6

7. Cop-win case consider the case when one cop has a winning strategy –cop-win graphs introduced by (Nowakowski, Winkler, 83), (Quilliot, 78) –cliques, universal vertices –trees –chordal graphs Random cop-win graphs Anthony Bonato 7

8. Characterization node u is a corner if there is a v such that N[v] contains N[u] –v is the parent; u is the child a graph is dismantlable if we can iteratively delete corners until there is only one vertex Theorem (Nowakowski, Winkler 83; Quilliot, 78) A graph is cop-win if and only if it is dismantlable. idea: cop-win graphs always have corners; retract corner and play shadow strategy; - dismantlable graphs are cop-win by induction Random cop-win graphs Anthony Bonato 8

9. Dismantlable graphs Random cop-win graphs Anthony Bonato 9

10. Dismantlable graphs Random cop-win graphs Anthony Bonato 10 unique corner! part of an infinite family that maximizes capture time (Bonato, Hahn, Golovach, Kratochvíl,09)

11. Cop-win orderings a permutation v 1, v 2, …, v n of V(G) is a cop-win ordering if there exist vertices w 1, w 2, …, w n such that for all i, w i is the parent of v i in the subgraph induced V(G) \ {v j : j < i}. –a cop-win ordering dismantlability Random cop-win graphs Anthony Bonato 11 1 2 3 4 5

12. Cop-win Strategy (Clarke, Nowakowski, 2001) V(G) = [n] a cop-win ordering G 1 = G, i > 1, G i : subgraph induced by deleting 1, …, i-1 f i : G i → G i+1 retraction mapping i to a fixed one of its parents F i = f i-1 ○… ○ f 2 ○ f 1 –a homomorphism idea: robber on u, think of F i (u) shadow of robber –cop moves to capture shadow –works as the F i are homomorphisms results in a capture in at most n moves of cop Random cop-win graphs Anthony Bonato 12

13. Random cop-win graphs Anthony Bonato 13 Random graphs G(n,p) (Erdős, Rényi, 63) n a positive integer, p = p(n) a real number in (0,1) G(n,p): probability space on graphs with nodes {1,…,n}, two nodes joined independently and with probability p 5 123 4

14. Typical cop-win graphs what is a random cop-win graph? G(n,1/2) and condition on being cop-win probability of choosing a cop-win graph on the uniform space of labeled graphs of ordered n Random cop-win graphs Anthony Bonato 14

15. Cop number of G(n,1/2) (B,Hahn, Wang, 07), (B,Prałat, Wang,09) A.a.s. c(G(n,1/2)) = (1+o(1))log 2 n. -matches the domination number Random cop-win graphs Anthony Bonato 15

16. Universal vertices P(cop-win) ≥ P(universal) = n2 -n+1 – O(n 2 2 -2n+3 ) = (1+o(1))n2 -n+1 …this is in fact the correct answer! Random cop-win graphs Anthony Bonato 16

17. Main result Theorem (B,Kemkes, Prałat,11+) In G(n,1/2), P(cop-win) = (1+o(1))n2 -n+1 Random cop-win graphs Anthony Bonato 17

18. Corollaries Corollary (BKP,11+) The number of labeled cop-win graphs is Random cop-win graphs Anthony Bonato 18

19. Corollaries U n = number of labeled graphs with a universal vertex C n = number of labeled cop-win graphs Corollary (BKP,11+) That is, almost all cop-win graphs contain a universal vertex. Random cop-win graphs Anthony Bonato 19

20. Strategy of proof probability of being cop-win and not having a universal vertex is very small 1.P(cop-win + ∆ ≤ n – 3) ≤ 2 -(1+ε)n 2.P(cop-win + ∆ = n – 2) = 2 -(3-log 2 3)n+o(n) Random cop-win graphs Anthony Bonato 20

21. P(cop-win + ∆ ≤ n – 3) ≤ 2 -(1+ε)n consider cases based on number of parents: a.there is a cop-win ordering whose vertices in their initial segments of length 0.05n have more than 17 parents. b.there is a cop-win ordering whose vertices in their initial segments of length 0.05n have at most 17 parents, each of which has co-degree more than n 2/3. c.there is a cop-win ordering whose initial segments of length 0.05n have between 2 and 17 parents, and at least one parent has co-degree at most n 2/3. d.there exists a vertex w with co-degree between 2 and n 2/3, such that w i = w for i ≤ 0.05n. Random cop-win graphs Anthony Bonato 21

22. P(cop-win + ∆ = n – 2) ≤ 2 -(3-log 2 3)n+o(n) Sketch of proof: Using (1), we obtain that there is an ε > 0 such that P(cop-win) ≤ P(cop-win and ∆ ≤ n-3) + P(∆ ≥ n-2) ≤ 2 -(1+ε)n + n 2 2 -n+1 ≤ 2 -n+o(n) (*) if ∆ = n-2, then G has a vertex w of degree n-2, a unique vertex v not adjacent to w. –let A be the vertices not adjacent to v (and adjacent to w) –let B be the vertices adjacent to v (and also to w) Claim: The subgraph induced by B is cop-win. Random cop-win graphs Anthony Bonato 22

23. Random cop-win graphs Anthony Bonato 23 A B w v x

24. Proof continued n choices for w; n-1 for v choices for A if |A| = i, then using (*), probability that B is cop-win is at most 2 -n+2+i+o(n) Random cop-win graphs Anthony Bonato 24

25. Problems do almost all k-cop-win graphs contain a dominating set of order k? –would imply that the number of labeled k-cop-win graphs of order n is –difficulty: no simple elimination ordering for k > 1 (Clarke, MacGillivray,09+) characterizing cop-win planar graphs (Clarke, Fitzpatrick, Hill, Nowakowski,10): classify the cop- win graphs which have cop number 2 after a vertex is deleted Random cop-win graphs Anthony Bonato 25

26. Random cop-win graphs Anthony Bonato 26 preprints, reprints, contact: Google: “Anthony Bonato”

27. Random cop-win graphs Anthony Bonato 27