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Almost all cop-win graphs contain a universal vertex

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

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Almost all cop-win graphs contain a universal vertex

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

  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

  3. Cops and Robbers • played on reflexive graphs G • two players CopsC 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

  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?” sfixed: running time O(n2s+3), n = |V(G)| • (Fomin, Golovach, Kratochvíl, Nisse, Suchan, 08): if s not fixed, then computing the cop number is NP-hard • (Shroeder,01) Ggenus g, then c(G)≤ ⌊ 3g/2 ⌋+3 • (Joret, Kamiński, Theis, 09) c(G)≤ tw(G)/2 Random cop-win graphs Anthony Bonato

  5. Meyniel’s Conjecture • c(n) = maximum cop number of a connected graph of order n • Meyniel Conjecture: c(n) = O(n1/2). Random cop-win graphs Anthony Bonato

  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(n1-ε) for some ε > 0 is open Random cop-win graphs Anthony Bonato

  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

  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

  9. Dismantlable graphs Random cop-win graphs Anthony Bonato

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

  11. Cop-win orderings • a permutation v1, v2, … , vnof V(G) is a cop-win ordering if there exist vertices w1, w2, …, wnsuch that for all i, wi is the parent of vi in the subgraph induced V(G) \ {vj : j < i}. • a cop-win ordering dismantlability 5 1 4 3 2 Random cop-win graphs Anthony Bonato

  12. Cop-win Strategy (Clarke, Nowakowski, 2001) • V(G) = [n] a cop-win ordering • G1 = G, i > 1, Gi: subgraph induced by deleting 1, …, i-1 • fi: Gi → Gi+1 retraction mapping i to a fixed one of its parents • Fi=fi-1 ○… ○ f2 ○ f1 • a homomorphism • idea: robber on u, think of Fi(u) shadow of robber • cop moves to capture shadow • works as the Fi are homomorphisms • results in a capture in at most n moves of cop 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 4 1 2 3 5 Random cop-win graphs Anthony Bonato

  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

  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))log2n. -matches the domination number Random cop-win graphs Anthony Bonato

  16. Universal vertices • P(cop-win) ≥ P(universal) =n2-n+1 – O(n22-2n+3) = (1+o(1))n2-n+1 • …this is in fact the correct answer! Random cop-win graphs Anthony Bonato

  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

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

  19. Corollaries Un= number of labeled graphs with a universal vertex Cn= 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

  20. Strategy of proof • probability of being cop-win and not having a universal vertex is very small • P(cop-win + ∆ ≤ n – 3) ≤ 2-(1+ε)n • P(cop-win + ∆ = n – 2) = 2-(3-log23)n+o(n) Random cop-win graphs Anthony Bonato

  21. P(cop-win + ∆ ≤ n – 3) ≤2-(1+ε)n • consider cases based on number of parents: • there is a cop-win ordering whose vertices in their initial segments of length 0.05n have more than 17 parents. • 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 n2/3. • 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 n2/3. • there exists a vertex w with co-degree between 2 and n2/3, such that wi = w for i ≤ 0.05n. Random cop-win graphs Anthony Bonato

  22. P(cop-win + ∆ = n – 2) ≤2-(3-log23)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 + n22-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

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

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

  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

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

  27. Random cop-win graphs Anthony Bonato

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