Discrete Mathematics 312 (2012) 1652–1657 Contents lists available at SciVerse ScienceDirect Discrete Mathematics journal homepage: www.elsevier.com/locate/disc Almost all cop-win graphs contain a universal vertex Anthony Bonato, Graeme Kemkes, Paweª Praªat ∗ Department of Mathematics, Ryerson University, Toronto, ON, M5B 2K3, Canada article info a b s t r a c t Article history: We consider cop-win graphs in the binomial random graph G.n; 1=2/. We prove that Received 23 August 2011 almost all cop-win graphs contain a universal vertex. From this result, we derive that Received in revised form 14 February 2012 the asymptotic number of labelled cop-win graphs of order n is equal to .1 C o.1// Accepted 15 February 2012 n2=2−3n=2C1 Available online 8 March 2012 n2 . ' 2012 Elsevier B.V. All rights reserved. Keywords: Cop-win graph Random graphs Universal vertex Cop-win ordering 1. Introduction Cops and Robbers is vertex-pursuit game played on a reflexive graph. There are two players, consisting of a set of cops and a single robber. The game is played over a sequence of discrete time-steps or rounds, with the cops going first in the first round and then playing alternate time-steps. The cops and the robber occupy vertices. When a player is ready to move in a round they must move to a neighbouring vertex. Because of the loops, players can pass, or remain on their own vertex. Observe that any subset of cops may move in a given round. The cops win if, after some finite number of rounds, one of them can occupy the same vertex as the robber. This is called a capture. The robber wins if he can evade capture indefinitely. A winning strategy for the cops is a set of rules that, if followed, result in a win for the cops. A winning strategy for the robber is defined analogously. If we place a cop at each vertex, then the cops are guaranteed to win. Therefore, the minimum number of cops required to win in a graph G is a well-defined positive integer, named the cop number (or copnumber) of the graph G. We write c.G/ for the cop number of a graph G. If c.G/ D k, then we say that G is k-cop-win. In the special case k D 1, we say that G is cop-win (or copwin). Nowakowski and Winkler [10], and independently Quilliot [13], considered the game with one cop only; the introduction of the cop number came in [1]. Many papers have now been written on cop number since these three early works; see the surveys [2,8,9]. Since their introduction, the structure of cop-win graphs has been relatively well understood. In [10,13,14], a kind of ordering of the vertex set – now called a cop-win or elimination ordering – was introduced which completely characterizes such graphs. If u is a vertex, then the closed neighbour set of u, written NTuU, consists of u along with the neighbours of u. A vertex u is a corner if there is some vertex v, v 6D u, such that NTuU ⊆ NTvU. We say that v is the parent of u, and that u is the child of v. A graph is dismantlable if some sequence of deleting corners results in the graph with a single vertex. For example, each tree is dismantlable, and, more generally, so are chordal graphs (that is, graphs with no induced cycles of length more than 3). To prove the latter fact, note that a chordal graph contains a vertex whose neighbour set is a clique; see, for example, West [16]. ∗ Corresponding author. E-mail addresses: [email protected] (A. Bonato), [email protected] (G. Kemkes), [email protected] (P. Praªat). 0012-365X/$ – see front matter ' 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.disc.2012.02.018 A. Bonato et al. / Discrete Mathematics 312 (2012) 1652–1657 1653 The following theorem gives the main results characterizing cop-win graphs. Theorem 1.1 ([10,13,14]). (1) If u is a corner of a graph G, then G is cop-win if and only if G − u is cop-win. (2) A graph is cop-win if and only if it is dismantlable. From Theorem 1.1, cop-win (or dismantlable) graphs have a recursive structure, made explicit in the following sense. A permutation v1; v2; : : : ; vn of the vertices of G is a cop-win ordering if there exist vertices w1; w2; : : : ; wn such that, for all i 2 TnU D f1; 2;:::; ng, NTviU ⊆ NTwiU in V .G/ n fvj V j < ig and vi 6D wi. We use the notation v for a cop-win ordering, and w for its parent sequence. Cop-win orderings are sometimes called elimination orderings, as we delete the vertices from lower to higher index until only vertex vn remains. We say that an event holds asymptotically almost surely (a.a.s.) if it holds with probability tending to 1 as n tends to infinity. The probability of an event A is denoted by P.A/: Our goal is to investigate the structure of random cop-win graphs. The random graph model we use is the familiar G.n; 1=2/ probability space of all labelled graphs on n vertices, where each pair of vertices is joined with probability 1/2, independently of the events for other pairs of vertices. Note that a given graph G on eG edges occurs with probability e n −e n 1 G 1 . 2 / G 1 . 2 / P.G 2 G.n; 1=2// D 1 − D ; 2 2 2 which does not depend on G. Thus, G.n; 1=2/ is in fact a uniform probability space over all labelled graphs on n vertices. We heavily use this interpretation of G.n; 1=2/ in the proof of our main result, Theorem 2.1, stated below. We expect results analogous to Theorem 2.1 (that is, with 2 replaced by 1=p) for other constants p 2 .0; 1/ and p D p.n/ tending to zero with n. (The argument for p D p.n/ tending to 1 needs to be modified when the expected number of universal vertices is Ω.1/; see [11].) However, this seems not to be an interesting research direction in the theory of random graphs, where we usually focus on investigating typical properties that hold a.a.s. in G.n; p/. Therefore, studying bounds for the cop number that hold a.a.s. are of interest, and a number of papers have been published on this topic (see, for example [3,6,15,12] and the recent monograph [5]). We focus on G.n; 1=2/ in this paper since it gives the typical structure of a cop-win graph. Therefore, from now on our probability space is always taken to be G.n; 1=2/. 2. Main results A vertex is universal if it is joined to all others. Let cop-win be the event that the graph is cop-win, and let universal be the event that there is a universal vertex. If a graph has a universal vertex w, then it is cop-win; in a certain sense, graphs with universal vertices are the simplest cop-win graphs. The probability that a random graph is cop-win can be estimated as follows: −nC1 2 −2nC3 P.cop-win/ ≥ P.universal/ D n2 − O.n 2 / − C D .1 C o.1//n2 n 1: (2.1) Surprisingly, this lower bound is in fact the correct asymptotic value for P.cop-win/. Our main result is the following theorem. Theorem 2.1. In G.n; 1=2/, we have that −nC1 P.cop-win/ D .1 C o.1//n2 : Using Theorem 2.1, we derive the asymptotic number of labelled cop-win graphs. Corollary 2.2. The number of cop-win graphs on n labelled vertices is n −nC1 n2=2−3n=2C1 .1 C o.1//2. 2 /n2 D .1 C o.1//n2 : It also follows that almost all cop-win graphs contain a universal vertex, a fact not obvious a priori. Corollary 2.3. P.universal j cop-win/ D 1 − o.1/: We prove Theorem 2.1 in the next section. We finish this section with some notation that will be used in the proof. The degree of a vertex u is written deg.u/. We let ∆.G/ denote the maximum degree of G (or just ∆ if G is clear from context). The co-degree of a vertex u in a graph of order n is n − 1 − deg.u/. 3. Proofs of main results To prove Theorem 2.1, we bound the probability of cop-win for graphs of maximum degree at most n−2. Since the proof for ∆ D n − 2 has a different flavour than the one for ∆ ≤ n − 3, we prove it independently. 1654 A. Bonato et al. / Discrete Mathematics 312 (2012) 1652–1657 Theorem 3.1. (a) For some ϵ > 0, we have that −.1Cϵ/n P.cop-win and ∆ ≤ n − 3/ ≤ 2 : −.3−log 3/nCo.n/ (b) P.cop-win and ∆ D n − 2/ ≤ 2 2 . Theorem 2.1 follows immediately from Theorem 3.1 and (2.1). Proof of Theorem 3.1(a). Let G be a random graph drawn from the G.n; 1=2/ distribution. We study the probability that ∆ ≤ n − 3, and that there exists a permutation v D .v1; v2; : : : ; vn/ and a sequence of vertices w D .w1; w2; : : : ; wn/ which are a cop-win ordering and associated parent sequence for G, respectively.