Almost all cop-win graphs contain a universal vertex Anthony Bonato Ryerson University Random cop-win graphs Anthony Bonato

Random cop-win graphs Anthony Bonato Cops and Robbers C C R C Random cop-win graphs Anthony Bonato

Random cop-win graphs Anthony Bonato Cops and Robbers C C R C Random cop-win graphs Anthony Bonato

Random cop-win graphs Anthony Bonato Cops and Robbers C R C C cop number c(G) ≤ 3 Random cop-win graphs Anthony Bonato

Random cop-win graphs Anthony Bonato 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) ≤ δ(G) Random cop-win graphs Anthony Bonato

Fast facts about cop number (Aigner, Fromme, 84) introduced parameter G planar, then c(G) ≤ 3 (Berrarducci, Intrigila, 93), (B, Chiniforooshan,10): “c(G) ≤ s?” s fixed: 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) G genus g, then c(G) ≤ ⌊ 3g/2 ⌋+3 (Joret, Kamiński, Theis, 09) c(G) ≤ tw(G)/2 Random cop-win graphs Anthony Bonato

Random cop-win graphs Anthony Bonato 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

Random cop-win graphs Anthony Bonato State-of-the-art (Lu, Peng, 09+) proved that independently proved by (Scott, Sudakov,10+), and (Frieze, Krivelevich, Loh, 10+) (Bollobás, Kun, Leader, 08+): if p = p(n) ≥ 2.1log n/ n, then c(G(n,p)) ≤ 160000n1/2log n (Prałat,Wormald,11+): removed log factor Random cop-win graphs Anthony Bonato

Random cop-win graphs Anthony Bonato 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

Random cop-win graphs Anthony Bonato 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

Dismantlable graphs 8/25/2019 Anthony Bonato

Dismantlable graphs unique corner! part of an infinite family that maximizes capture time (Bonato, Hahn, Golovach, Kratochvíl,09) 8/25/2019 Anthony Bonato

Random cop-win graphs Anthony Bonato Cop-win orderings a permutation v1, v2, … , vn of V(G) is a cop-win ordering if there exist vertices w1, w2, …, wn such 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

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

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

Random cop-win graphs Anthony Bonato 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

Random cop-win graphs Anthony Bonato 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

Random cop-win graphs Anthony Bonato 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

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

Random cop-win graphs Anthony Bonato 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

Random cop-win graphs Anthony Bonato 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

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

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

Random cop-win graphs Anthony Bonato x v Random cop-win graphs Anthony Bonato

Random cop-win graphs Anthony Bonato 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

Random cop-win graphs Anthony Bonato Problems higher cop number? 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 or outer-planar graphs Random cop-win graphs Anthony Bonato

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

Random cop-win graphs Anthony Bonato

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