Chasing a Fast Robber on Planar Graphs and Random Graphs

Transcription

1 Chasing a Fast Robber on Planar Graphs an Ranom Graphs Noga Alon Abbas Mehrabian Abstract We consier a variant of the Cops an Robber game, in which the robber has unboune spee, i.e., can take any path from her vertex in her turn, but she is not allowe to pass through a vertex occupie by a cop. Let c (G) enote the number of cops neee to capture the robber in a graph G in this variant, an let tw(g) enote the treewith of G. We show that if G is planar then c (G) = Θ(tw(G)), an there is a constant-factor approximation algorithm for computing c (G). We also etermine, up to constant factors, the value of c (G) of the Erős-Rényi ranom graph G = G(n, p) for all amissible values of p, an show that when the average egree is ω(1), c (G) is typically asymptotic to the omination number. Keywors: Cops an Robber game, Fast robber, Planar graphs, Treewith, Ranom graphs, Domination number 1 Introuction The game of Cops an Robber is a perfect information game, playe in a graph G. The players are a set of cops an a robber. Initially, the cops are place at vertices of their choice in G (where more than one cop can be place at a vertex). Then the robber, being fully aware of the cops placement, positions herself at one of the vertices of G. Then the cops an the robber move in alternate rouns, with the cops moving first; however, Sackler School of Mathematics an Blavatnik School of Computer Science, Tel Aviv University, Tel Aviv 69978, Israel. nogaa@tau.ac.il. Research supporte in part by an ERC Avance grant, by a USA-Israeli BSF grant an by the Israeli I-Core program. Department of Combinatorics an Optimization, University of Waterloo, Waterloo ON, Canaa. amehrabi@uwaterloo.ca 1

2 players are permitte to remain stationary in their turn if they wish. The players use the eges of G to move from vertex to vertex. The cops win, an the game ens, if eventually a cop moves to the vertex currently occupie by the robber; otherwise, that is, if the robber can elue the cops forever, the robber wins. This game was efine (for one cop) by Winkler an Nowakowski [28] an Quilliot [32], an has been stuie extensively, see Hahn [22] or Bonato an Nowakowski [8]. The best known open question in this area is Meyniel s conjecture, publishe by Frankl [19], which states that for every connecte graph on n vertices, O( n) cops are sufficient to capture the robber. One intriguing fact about the Cops an Robber game is that although many scholars have stuie the game it is not yet well unerstoo. In particular, although the upper boun O( n) was conjecture alreay in 1987, no upper boun better than n 1 o(1) has been prove since then (see [20, 24, 33]). One might try to change the rules of the game slightly in orer to get a more approachable problem, an/or to unerstan what property of the original game causes the ifficulty. Thus various variations of the game have been stuie [7, 11, 20, 23, 2]. The approach chosen by Fomin, Golovach, Kratochvíl, Nisse, an Suchan [18] is to allow the robber move faster than the cops. Inspire by their work, in this paper we let the robber take any path from her current position in her turn, but she is not allowe to pass through a vertex occupie by a cop. The parameter of interest is the cop number of G, which is efine as the minimum number of cops neee to ensure that the cops can win. We enote the cop number of G by c (G), in which the at the subscript inicates that the robber has unboune spee. This variant was first stuie by Fomin, Golovach, Kratochvíl [17]. They prove that computing c (G) is an NP-har problem, even if G is a split graph. (A split graph is a graph whose vertex set can be partitione into a clique an an inepenent set.) Next Gavenčiak [21] gave a polynomial time algorithm for interval graphs. This variant was further stuie by Frieze, Krivelevich an Loh [20], where the authors approach is base on expansion. In [20], it is shown that for each n, there exists a connecte graph on n vertices with cop number Θ(n). See the thesis of the secon author [27] for more results on this variant, in particular about graphs with cop number one, interval graphs, choral graphs, expaner graphs an Cartesian proucts of graphs. We stuy this game on planar graphs an ranom graphs. Let tw(g) enote the treewith of the graph G (see the next section for the formal efinition). For planar graphs, we prove the following. 2

3 Theorem 1. Let G be a connecte planar graph on n vertices. Then we have c (G) = Θ(tw(G)) = O( n), an there is a constant-factor approximation algorithm for computing c (G). In fact, we show that the conclusion of Theorem 1 is true for graphs G that o not have a fixe apex graph as a minor. (An apex graph is a graph H that has a vertex v such that H v is planar.) Note that the m m gri has cop number Ω(m) (see Theorem 3.3) so the boun c (G) = O( n) in Theorem 1 is tight. We enote the Erős-Rényi ranom graph with parameters n an p by G(n, p). All asymptotics throughout are as n. We say that an event in a probability space hols asymptotically almost surely (a.a.s.) if the probability that it hols approaches 1 as n goes to infinity. The secon author [26] showe that if np 4.2 log n, then there are positive constants β 1, β 2 such that a.a.s., β 1 /p c (G(n, p)) β 2 log(np)/p. Let γ(g) enote the omination number of the graph G (the formal efinition appears in the next section). We prove the following theorem, tightening the result above an extening it to all amissible values of p. Theorem 2. (a) If 27 np = O(1), then there exist positive constants η 1, η 2 such that a.a.s. η 1 log(np)/p c (G) γ(g) η 2 log(np)/p. (b) If np = ω(1) an p = 1 Ω(1), then a.a.s. log(np) c (G) = (1 + o(1)) log(1 p). (c) If np = ω(1), then a.a.s. c (G) = (1 + o(1))γ(g). Note that if np < 27 then a.a.s. the graph has Ω(n) isolate vertices, an hence in this case c (G) = Θ(n). Therefore, the above theorem an the fact that the proof of its last part presente in Section 4 shows that for all Ω(1) p 1, c (G) = γ(g) a.a.s. etermines the typical asymptotic behavior of c (G(n, p)) for all amissible value of p. The lollipop graph L n is obtaine from a complete graph on n vertices an a path on n + 1 vertices by ientifying some vertex of the complete graph with an en vertex of the 3

4 path. Notice that c (H) γ(h) is true for any graph H, since if the cops start from a ominating set, they will capture the robber in the first roun. Also, it is not har to see that c (H) tw(h) + 1 is true for any graph H (see Theorem 3.3). These upper bouns are far from being tight for general graphs, as L n has cop number one but omination number an treewith Θ(n). Theorems 1 an 2 state that the two crue upper bouns are actually tight up to constant factors for two important classes of graphs. 2 Preliminaries Let G be the graph in which the game is playe. In this paper G is always finite, an n always enotes the number of vertices of G. We will assume that G is simple, because eleting multiple eges or loops oes not affect the set of possible moves of the players. Note that the cop number of a isconnecte graph equals the sum of the cop numbers of its connecte components, an hence it suffices to unerstan the behavior of this parameter for connecte graphs. As we are only intereste in stuying the cop number (an not the number of rouns in the game) we may assume without loss of generality that the cops choose vertices of our choice in the beginning, since they can move to the vertices of their choice later. For a subset A of vertices, the neighbourhoo of A, enote by N(A), is the set of vertices in V (G) \ A that have a neighbour in A, an the close neighbourhoo of A, written N(A), is the union A N(A). If A = {v} then we may write N(v) an N(v) instea of N(A) an N(A), respectively. A ominating set is a subset A of vertices with V (G) = N(A), an the omination number of G, written γ(g), is the minimum size of a ominating set of G. The subgraph inuce by A is written G[A], an the subgraph inuce by V (G) \ A is written G A. All logarithms are in the natural base. Write = (G) for the maximum egree in G. A tree ecomposition of a graph G is a pair (T, W ), where T is a tree an W = (W t : t V (T )) is a family of subsets of V (G) such that (i) t V (T ) W t = V (G), (ii) every ege of G has both enpoints in some W t, an (iii) For every v V (G), the set {t : v W t } inuces a subtree of T. 4

5 The with of (T, W ) is max{ W t 1 : t V (T )}, an the treewith of G, written tw(g), is the minimum with of a tree ecomposition of G. We will use the following facts about tree ecompositions, whose proofs can be foun in Section 12.3 of [15]. Proposition 2.1. Let (T, W ) be a tree ecomposition of a graph G. (a) Let A be the vertex set of a clique in G. Then there is a t V (T ) with A W t. (b) Let t 1 t 2 be an ege of T, an let T 1 an T 2 be the components of T t 1 t 2, with t 1 T 1 an t 2 T 2. Define X = W t1 W t2, U 1 = t T1 W t an U 2 = t T2 W t. Then X is a cut-set in G, an there is no ege between U 1 \ X an U 2 \ X. We will use the following large eviations inequalities (see, e.g., Appenix A of [4]). Proposition 2.2. Let X = X 1 + X X m, where the X i are inepenent ranom variables taking values in {0, 1}. We have the following inequalities. (a) (b) P[X E[X] t], P[X E[X] + t] exp( 2t 2 /m) 0 t. P[X (1 ɛ)e[x]] exp( ɛ 2 E[X]/2) 0 ɛ. 3 Planar Graphs In one of the first papers on the original Cops an Robber game, Aigner an Fromme [1] prove that three cops can capture the robber in any planar graph. In this section, we prove Theorem 1 that eals with the case of a fast robber in a planar graph. Here is a high-level sketch of the proof. First, by relating our Cops an Robber game with the so-calle Helicopter Cops an Robber game of Seymour an Thomas [34], we show that for any graph G, tw(g) + 1 (G) + 1 c (G) tw(g)

6 Next, since the cop number cannot increase by contracting the eges, we may use a theorem from the biimensionality theory of Demaine an Hajiaghayi [13] to infer that c (G) = Ω(tw(G)). The Helicopter Cops an Robber game has two versions, an the one we efine here is calle jump-searching. Definition (Helicopter Cops an Robber game (the jump-searching version)). For X V (G), an X-flap is the vertex set of a connecte component of G X. Two subsets X, Y V (G) touch if N(X) Y. A position is a pair (X, R), where X V (G) an R is an X-flap. (In the game X is the set of vertices currently occupie by the cops an R tells us where the robber is since she can run arbitrarily fast, all that matters is which component of G X contains her.) At the start, the cops choose a subset X 0, an the robber chooses an X 0 -flap R 0. Note that if there are k cops in the game, then X 0 k. At the start of roun i, we have some position (X i 1, R i 1 ). The cops choose a new set X i V (G) with X i k (an no other restriction), an announce it. Then the robber, knowing X i, chooses an X i -flap R i which touches R i 1. If this is not possible then the cops have won. Otherwise, i.e. if the robber never runs out of vali moves, the robber wins. The following lemma establishes a link between the two games. Lemma 3.1. Let G be a graph. If k cops can capture a robber with unboune spee in the Cops an Robber game in G, then k( + 1) cops can capture the robber in the Helicopter Cops an Robber game in G. Proof. We consier two games playe in two copies of G: the first one, which we call the real game, is a game of Helicopter Cops an Robber with k( + 1) cops; an the secon one, the virtual game, is the usual Cops an Robber game with k cops an a robber with unboune spee. Given a winning strategy for the cops in the virtual game, we nee to give a capturing strategy for the cops in the real game. We translate the moves of the cops from the virtual game to the real game, an translate the moves of the robber from the real game to the virtual game, in such a way that all the translate moves are vali, an if the robber is capture in the virtual game, then she is capture in the real game as well. Hence, as the cops have a winning strategy in the virtual game, they have a winning strategy in the real game, too. In the virtual game, initially the cops choose a subset C 0 of vertices. Then the real cops choose X 0 = N(C 0 ). Recall that C 0 k, so X 0 k( + 1). The real robber chooses 6

7 R 0, which is an X 0 -flap, an the virtual robber chooses an arbitrary vertex r 0 R 0. In general, at the en of roun i 1 we have X i 1 = N(C i 1 ) an r i 1 R i 1. Suppose the virtual robber is not capture in roun i. In roun i, first the virtual cops move to a new set C i. Each cop either stays still or moves to a neighbour, thus C i N(C i 1 ) = X i 1 an since R i 1 was an X i 1 -flap, C i R i 1 =. The real cops choose X i = N(C i ) an announce it. The real robber, knowing X i, chooses an X i -flap R i that touches R i 1. If she cannot fin a vali move then she is capture an the lemma is prove. Otherwise, note that by efinition C i R i =. Let r i be an arbitrary vertex of R i. The virtual robber moves from r i 1 to r i. Since R i 1 an R i touch, an both of them are connecte, R i 1 R i is connecte. Moreover, C i oes not intersect R i 1 R i, so this is a vali move in the virtual game. Now, suppose the virtual robber is capture in roun i. We claim that if this happens then the real robber has alreay been capture in one of the previous rouns. If this is not the case, then in roun i, the virtual cops move to a new set C i such that r i 1 C i. Each cop either stays still or moves to a neighbour, thus C i N(C i 1 ) = X i 1 an since R i 1 was an X i 1 -flap, C i R i 1 =. But r i 1 C i because the virtual robber has been capture in roun i, an r i 1 R i 1, thus r i 1 C i R i 1, which is a contraiction. This shows that the real robber will be capture even before the virtual robber, an the proof is complete. Seymour an Thomas [34] prove the following theorem. Theorem 3.2 ([34]). The minimum number of cops neee to capture a robber in Helicopter Cops an Robber game is equal to the treewith of the graph plus one. Using this we have the following. Theorem 3.3. For every graph G we have tw(g) + 1 (G) + 1 c (G) tw(g) + 1. Proof. The lower boun follows from Lemma 3.1 an Theorem 3.2. To prove the upper boun, consier a tree ecomposition (T, W ) of G having minimum with. Assume that there are tw(g) + 1 cops in the game, so for every t V (T ), there are at least W t cops in the game. The cops start at W t1 for some arbitrary t 1 V (T ). Assume that the robber starts at r 0, an let t be such that r 0 W t. Let t 2 be the neighbour of t 1 in the unique (t 1, t)-path in T. Let T 1 an T 2 be the components of T t 1 t 2, with t 1 T 1 an 7

8 t 2 T 2. Define X = W t1 W t2, U 1 = t T1 W t, an U 2 = t T2 W t. So the cops are all in U 1 an the robber is at a vertex in U 2 \ X. Note that the number of cops is at least W t2. Now the cops move in orer to occupy W t2, in such a way that the cops in X stay still. After some rouns, the cops will be locate at W t2, an uring those rouns the robber coul not escape from U 2 \ X, because by part (b) of Proposition 2.1, there is no ege between U 1 \ X an U 2 \ X. When the cops have covere W t2, the total space available to the robber has been ecrease. Continuing similarly the cops will eventually capture the robber. Remark. The complete graph on n vertices shows that the lower boun is tight. The upper boun is also tight: start with m 4 vertices, an a m isjoint paths of length 3 between any two of them. This graph has treewith m 1 an cop number m. The etails can be foun in Theorem 4.5 of [27]. Recall that an apex graph is a graph H that has a vertex v such that H v is planar. The following theorem was prove in a weaker form by Demaine, Fomin, Hajiaghayi, an Thilikos [12], an then in its current form by Demaine an Hajiaghayi [14]. Theorem 3.4 ([12, 14]). Let H be a fixe apex graph. There is a constant C H such that the following hols. Let g : N N be a strictly increasing function, an P (G) be a graph parameter with the following two properties. 1. If G is the r r gri augmente with aitional eges such that each vertex is incient to C H eges connecte to non-bounary vertices of the gri, then P (G) g(r). 2. P (G) oes not increase by contracting an ege of G. Then, for any graph G that oes not contain H as a minor, the treewith of G is O (g 1 (P (G))). Now, we prove Theorem 1. Proof of Theorem 1. We show that the parameter c (G) satisfies the two properties given in Theorem 3.4, with g(r) = (r+1)/(5+c H ). First, an augmente r r gri has treewith r an maximum egree at most 4 + C H, so by Theorem 3.3 its cop number is at least (r + 1)/(5 + C H ). Secon, we nee to show that the cop number oes not increase by contracting an ege. It is easy to see that contracting an ege oes not help the robber, since she has unboune spee, an it oes not hurt the cops. Therefore, contracting an ege oes not increase the cop number. 8

9 Let H be the complete graph on 5 vertices. Since G is planar, it oes not contain H as a minor. Therefore, by Theorem 3.4, we have tw(g) = O(c (G)). By Theorem 3.3, c (G) tw(g) + 1, so we have c (G) = Θ(tw(G)). Moreover, it is known (see, e.g., [3]) that any G that oes not have H as a minor has tw(g) = O( n). Finally, Feige, Hajiaghayi, an Lee [16] have evelope an O(1)-approximation algorithm for fining the treewith of a graph that oes not contain H as a minor. 4 Ranom Graphs The original Cops an Robber game for Erős-Rényi ranom graphs has been stuie by several authors [6, 9, 25, 30]. In particular, Pra lat an Wormal [31] prove that Meyniel s conjecture hols for ranom graphs. In this section, we prove Theorem 2 that etermines the typical asymptotic behavior of c (G) for the Erős-Rényi ranom graph. Here is a high-level sketch of the proof. To give an upper boun for the cop number, we simply boun the omination number. For the lower boun, we give an escaping strategy for the robber. We first fin a number s such that a.a.s. there is an ege between any two isjoint subsets of vertices of size s. It follows that any subset A of vertices of size 3s inuces a connecte component of size s. Call this component a continent of A. Now, if the number of cops is small enough such that when they are in a subset S of vertices, N(S) leaves out at least 3s vertices, then the robber always moves to a continent R of V (G) \ N(S). Assume that the cops move to a new subset S in their turn, so N(S ) also leaves out at least 3s vertices, hence there is a continent R V (G) \ N(S ). Now, since R an R have size at least s, if they o not intersect, there is an ege between them. In either case the robber can move to R, an thus she will never be capture. Let G = G(n, p) an = (n) := np. As mentione after the statement of Theorem 2, the fact that for, say, < 27, c (G) = Θ(n) a.a.s. is very simple, an hence we may an will assume that is bigger. In particular, assume that > exp(1), an that n is sufficiently large. Lemma 4.1. A.a.s. for any two isjoint subsets A 1, A 2 V (G) of size at least en log /, there exists an ege between A 1 an A 2. Proof. By the union boun, the probability that there exist isjoint A 1, A 2 of size en log / 9

10 with no ege between them is at most ( ) 2 n en log (1 p) ( en log ( ) 2en log ) 2 e e pe2 n 2 log 2 e log 2 ( 2en log (log log log ) e 2 n log 2 ) = exp = o(1). The following lemma can be easily prove (see, e.g., Lemma 7.3 in [27]). Lemma 4.2. Let a 1, a 2,..., a m be positive integers such that each of them is at most t/2, an their sum is t. Then one can choose a subset of {a 1,..., a m } whose sum is between t/3 an t/2 (inclusive). The following corollary follows from Lemmas 4.1 an 4.2. Corollary 4.3. A.a.s. for any A V (G) of size at least 3en log /, there exists a connecte component of G[A] of size at least en log /. We are now reay to prove our main lower boun for the cop number. Theorem 4.4. Assume that p = o(1). Let α (0, 1) be such that for sufficiently large n, an Then a.a.s. α (3e + α) log > 0, ( 2 α (3e + α) log ) 2 + α log (1 + log log α log log ) < 0. c (G) > αn log /. Remark. For any with > 3e log such an α exists. The largest for which such an α oes not exist is aroun Proof. Claim. A.a.s. for any subset X V (G) of size αn log /, we have V (G) \ N(X) > (3e + α)n log /. 10

11 Proof of Claim. Let X be a set of size αn log /. Let α > α be small enough so that it also satisfies an α (3e + α) log > 0, (1) ( 2 α (3e + α) log ) 2 + α log (1 + log log α log log ) < 0, (2) an set y = α log. Then for any vertex v V (G), since p = o(1), for n large enough, P[v / N(X)] (1 p) X exp( pα /α) X = e y. Thus E[ V (G) \ N(X) ] ne y = n α. Note that by (1), so by Proposition 2.2(a), n α > P[ V (G) \ N(X) (3e + α)n log /] exp The total number of such sets X is ( ) ( ) αn log n e αn log α log By (2) we have [ exp (3e + α)n log, = exp [ [ 2 2 ( n α ( α ( αn log (1 + log log α log log ) 2 α ) 2 (3e + α)n log /n] ) 2 (3e + α) log n]. ( ) αn log = exp (1 + log log α log log ), so by the union boun, a.a.s. for every X of size αn log / we have V (G) \ N(X) > (3e + α)n log /. ) 2 (3e + α) log n] = o(1), Assuming that there are αn log / cops in the game, we give an escaping strategy for the robber. For a subset S V (G), enote by G/S the subgraph obtaine by eleting the vertices N(S). The following conitions are satisfie a.a.s. 11

12 1. For any subset S V (G) of size at most αn log /, G/S has at least 3en log / vertices. This is true with probability 1 o(1) by the claim. 2. For any subset S V (G) of size at most αn log /, the largest connecte component of G/S has at least en log / vertices. This is true with probability 1 o(1) by Corollary 4.3 an the previous conition. 3. For any two isjoint subsets A 1, A 2 V (G) of size at least e log(np)/p, there is an ege between A 1 an A 2. This is true with probability 1 o(1) by Lemma 4.1. The robber plays in such a way that whenever the cops are in a subset S, she is in a component of G/S with at least en log / vertices. This insures that she will never be capture. She can clearly position herself as require in the beginning. Assume that at the en of roun i, the cops are in S i, the robber is in some vertex in C i, where C i is a component of G/S i with at least en log / vertices. In roun i + 1, the cops move to S i+1. Note that S i+1 an C i are isjoint. Let C i+1 be a component of G/S i+1 with at least en log / vertices. Either C i an C i+1 have a vertex in common, or they are isjoint. In the latter case, since both have size at least en log /, there is an ege between them. In either case the robber moves to a vertex in C i+1, an this completes the proof. Now we turn to proving upper bouns for γ(g), which results in upper bouns for c (G). Lemma 4.5. Let 0 be a positive integer an let H be a graph on n vertices, which has at most a vertices of egree less than 0 1. Then we have γ(h) 1 + log 0 0 n + a. Proof. Let q = log 0 / 0, an let A be the set of vertices of H with egree less than 0 1. Form a ranom subset X V (H) by choosing each vertex inepenently with probability q. Let Y X = V (H) \ (N(X) A), an note that X Y X A is a ominating set for H. For every vertex v, the probability that v Y X is at most (1 q) 0 e q 0, since for this to happen v must have egree at least 0 1, an none of v an its neighbors must have been chosen. We conclue that E[ X Y X A ] E[ X ] + E[ Y X ] + E[ A ] nq + ne q 0 + a 1 + log 0 0 n + a. 12

13 Hence there is a choice of X for which X Y X A 1 + log 0 0 n + a. By bouning the number of vertices having small egree, we get the following. Theorem 4.6. Let δ (0, 1) be fixe. (a) If = ω(1), then a.a.s. (b) If = O(1), then a.a.s. γ(g) 1 + log (1 δ)p + n exp( δ2 /2). γ(g) 1 + log (1 δ)p + (1 + o(1))n exp( δ2 /8). Proof. Let 0 = (1 δ). Say vertex v is terrible if its egree is less than 0 1. We show that if = ω(1), then the number of terrible vertices is a.a.s. at most n exp( δ 2 /2), an if = O(1) then the number of terrible vertices is a.a.s. at most (1 + o(1))n exp( δ 2 /8). Then we are one by Lemma 4.5 since 1 + log 0 0 n = 1 + log((1 δ)) (1 δ) n 1 + log (1 δ)p. (a) For every vertex v, eg(v) is a binomial ranom variable with parameters n 1 an p, so by Proposition 2.2(b), P[eg(v) 0 1] P[eg(v) (1 δ)e[eg(v)]] exp( δ 2 (n 1)p/2). Thus the expecte number of terrible vertices is at most n exp( δ 2 (n 1)p/2). Since = ω(1), by the Markov inequality the number of terrible vertices is a.a.s. at most n exp( δ 2 /2). (b) First, assume that n is an integer an let k = n. Partition V (G) arbitrarily into k parts B 1,..., B k of size k. For a vertex v in part B i, say v is ba if N(v) (V (G) \ B i ) < (1 δ)np 1. Clearly, any terrible vertex is ba. Observe that for any s vertices v 1,..., v s in the same part, the events {v j is ba : 1 j s} are mutually inepenent. Fix 13

14 an ɛ (0, 1) an we will show that a.a.s. the number of ba vertices is at most (1 + ɛ)n exp( δ 2 np/8). Say a part is ba if it has more than (1 + ɛ)k exp( δ 2 np/8) ba vertices. Consier a part B i = {v 1, v 2,..., v k } an let X j be the inicator variable for v j being ba. Thus X := X 1 + X X k is the number of ba vertices of B i. For each j, notice that E[ N(v j ) (V (G) \ B i ) ] = (n k)p so by Proposition 2.2(b) E[X j ] = P[ N(v j ) (V (G) \ B i ) < (1 δ)np 1] [ ( P N(v j ) (V (G) \ B i ) 1 δ ) ] (n k)p 2 exp( δ 2 (n k)p/4). Hence E[X] = E[X 1 + X X k ] k exp( δ 2 (n k)p/4). Let µ := k exp( δ 2 (n k)p/4). The probability that B i is ba is at most P[X > (1 + ɛ)k exp( δ 2 np/8)] P[X > (1 + ɛ)µ] exp( 2(ɛµ) 2 /k), where we have use Proposition 2.2(a) an the fact that the X i s are mutually inepenent. So by the union boun, the probability that there exists a ba part is at most k exp( 2(ɛµ) 2 /k) = k exp [ 2ɛ 2 k exp( δ 2 (n k)p/2) ], which is o(1) as np = O(1) an k = ω(1). Consequently, a.a.s. there is no ba part, an the total number of ba vertices is at most (1 + ɛ)n exp( δ 2 np/8). Now, assume that n is not an integer, an write V (G) = V 0 A, where V 0 is a square number, an A = O( n). Partitioning V 0 an oing the same analysis as above shows that there are at most (1 + o(1))n exp( δ 2 np/8) terrible vertices in V 0. Thus G has at most (1 + o(1))n exp( δ 2 np/8) + A = (1 + o(1))n exp( δ 2 np/8) terrible vertices, an the proof is complete. Finally, we prove Theorem 2. Proof of Theorem 2. (a) Since 3e log(27) < 27, there exists α > 0 satisfying the conitions of Theorem 4.4. Setting η 1 = α proves the lower boun. 14

15 For the upper boun, let δ = 8/27 an let η 2 be the constant satisfying 1 + log 27 1 δ exp( δ 2 27/8) = η 2 log 27. It can be verifie, by ifferentiating both sies with respect to x, that for all x 27 we have 1 + log x 1 δ Thus, by Theorem 4.6(b), a.a.s. + 2x exp( δ 2 x/8) η 2 log x. γ(g) 1 + log (1 δ)p + (1 + o(1))n exp( δ2 /8) η 2 n log /. (b) When p = Ω(1), the boun follows from part (c) of Theorem 4.6 in [26]. p = o(1), note that p = (1 + o(1))( log(1 p)). Fix ɛ > 0, an we will show that When (1 ɛ) log p c (G) γ(g) (1 + ɛ) log p. First, since = ω(1), α = 1 ɛ satisfies the conitions of Theorem 4.4, thus by that theorem, a.a.s. c (G) (1 ɛ)n log / = (1 ɛ) log p. For the upper boun, pick a δ (0, 1/2) small enough so that (1 δ)(1 + ɛ/2) 1. By Theorem 4.6(a), a.a.s. we have γ(g) 1 + log (1 δ)p + n exp( δ2 /2) = 1 [ 1 p 1 δ + log 1 δ + 2 exp ( δ 2 /2 ) ] 1 [ ( ɛ ) ] log + o(1) p 2 < (1 + ɛ) log /p. (c) When p = o(1), the proof of part (b) oes the job. Now, assume that p = Ω(1). Note that for this value of p, G is a.a.s. ( n)-connecte, an a.a.s. γ(g) < n. Inee, a.a.s. any two vertices of G have Ω(n) common neighbors, an hence even after eleting n vertices the remaining graph still has iameter 2 (in fact, it is known that a.a.s. the connectivity of G is equal to its minimum egree-see, e.g., [5] Chapter VII.2, but we only nee here the much weaker boun state above). Conitione on 15

16 these two events, we show that c (G) γ(g). Inee, if there are less than γ(g) cops in the game, then there exists a non-ominate vertex in every roun, an since G is γ(g)-connecte, there exists an unblocke path from the robber s vertex to that vertex; so the robber can move there, an will never be capture. 5 Concluing Remarks Note that each cop has two functions: attacking an blocking. Consier a version in which each cop can just attack, or in other wors, the robber can jump over the cops. Let c A (G) enote the cop number of G in this version. Now, consier another version in which each cop can just block, an say the robber is capture if she is in a vertex v such that the cops occupy N(v). Let c B (G) enote the cop number of G in the secon version. Then it is clear that c (G) c A (G) = γ(g), an that c (G) c B (G) tw(g) + 1. Therefore, in this notation Theorem 1 asserts that if G is planar, then c (G) = Θ(c B (G)). In other wors, blocking is the crucial function in planar graphs. On the other han, Theorem 2 asserts that if G is ranom with average egree ω(1), then c (G) c A (G), i.e., in a ranom graph, the attacking function is the crucial one. This shows an interesting contrast between planar graphs an ranom graphs in the context of pursuitevasion games. In the proof of Theorem 1, we showe that the cop number is a biimensional parameter, an use this to conclue that it is of the same orer as the treewith (in planar graphs). In almost all biimensional parameters stuie before [13], the parameter of interest is of quaratic orer in the treewith (a typical example is the omination number). So, this paper might be the first place where this theory is use to give a nontrivial result for a parameter which is linear in the treewith. We conclue with some open questions an irections for further research. As mentione in the introuction, treewith an omination number are two easy upper bouns for the cop number, an Theorems 1 an 2 imply that they are tight up to constant factors for two well stuie graph classes. It woul be interesting to fin other natural graph classes for which these bouns are tight. Let H be a fixe apex graph. It follows from the proof of Theorem 1 that if G oes not have H as a minor, then c (G) = Θ(tw(G)). It is natural to ask whether the conclusion is true when H is a general graph. Also, one can ask what is the largest constant κ such 16

17 that any planar G has κ tw(g) c (G). It woul be interesting to etermine the cop number of the m-imensional hypercube graph H m. Note that since H m has maximum egree m, treewith Θ (2 m / m) (see [10]), an omination number Θ (2 m /m) (see [29], or note that this follows easily from the existence of Hamming coes), there exist positive constants ζ 1, ζ 2 such that ζ 1 2 m m m c (H m ) ζ 22 m m. Fomin et al. [17] prove that computing c (G) is NP-har, but it is actually not known if this problem is in NP. To show that this problem is in NP, one nees to prove that there is always an efficient way to escribe the cops strategy. This has been one for the Helicopter Cops an Robber game [34]. As another algorithmic question, it woul be interesting to exten the constant-factor approximation algorithm of Theorem 1 for computing the cop number of planar graphs to other graph classes, an/or to prove harness of approximation results for general graphs. Acknowlegement. support an lots of fruitful iscussions. The secon author is grateful to Nick Wormal for continuous References [1] M. Aigner an M. Fromme, A game of cops an robbers, Discrete Appl. Math. 8 (1984), no. 1, MR (85f:90124) [2] N. Alon an A. Mehrabian, On a generalization of Meyniel s conjecture on the Cops an Robbers game, Electron. J. Combin. 18 (2011), no. 1, Paper 19, 7. MR (2012c:05205) [3] N. Alon, P. Seymour, an R. Thomas, A separator theorem for nonplanar graphs, J. Amer. Math. Soc. 3 (1990), no. 4, MR (91h:05076) [4] N. Alon an J. H. Spencer, The probabilistic metho, thir e., Wiley-Interscience Series in Discrete Mathematics an Optimization, John Wiley & Sons Inc., Hoboken, NJ, 2008, With an appenix on the life an work of Paul Erős. MR (2009j:60004) [5] B. Bollobás, Ranom graphs, Acaemic Press Inc. [Harcourt Brace Jovanovich Publishers], Lonon, MR (87f:05152) 17

18 [6] B. Bollobás, G. Kun, an I. Leaer, Cops an robbers in a ranom graph, arxiv: v1 [math.co]. [7] A. Bonato, E. Chiniforooshan, an P. Pra lat, Cops an Robbers from a istance, Theoret. Comput. Sci. 411 (2010), no. 43, MR (2012:05251) [8] A. Bonato an R. J. Nowakowski, The game of cops an robbers on graphs, Stuent Mathematical Library, vol. 61, American Mathematical Society, Provience, RI, MR [9] A. Bonato, P. Pra lat, an C. Wang, Pursuit-evasion in moels of complex networks, Internet Math. 4 (2007), no. 4, MR [10] L. Sunil Chanran an T. Kavitha, The treewith an pathwith of hypercubes, Discrete Math. 306 (2006), no. 3, MR (2006k:68135) [11] N. E. Clarke, A witness version of the cops an robber game, Discrete Math. 309 (2009), no. 10, MR (2011a:91080) [12] E. D. Demaine, F. V. Fomin, M. Hajiaghayi, an D. M. Thilikos, Biimensional parameters an local treewith, SIAM J. Discrete Math. 18 (2004/05), no. 3, (electronic). MR (2006c:05128) [13] E. D. Demaine an M. Hajiaghayi, The biimensionality theory an its algorithmic applications, The Computer Journal 51 (2008), no. 3, [14], Linearity of gri minors in treewith with applications through biimensionality, Combinatorica 28 (2008), no. 1, MR (2009g:05159) [15] R. Diestel, Graph theory, thir e., Grauate Texts in Mathematics, vol. 173, Springer-Verlag, Berlin, MR (2006e:05001) [16] U. Feige, M. Hajiaghayi, an J. R. Lee, Improve approximation algorithms for minimum weight vertex separators, SIAM J. Comput. 38 (2008), no. 2, MR (2009g:68267) [17] F. V. Fomin, P. A. Golovach, an J. Kratochvíl, On tractability of cops an robbers game, Fifth IFIP International Conference on Theoretical Computer Science TCS 2008, IFIP Int. Fe. Inf. Process., vol. 273, Springer, New York, 2008, pp MR (2012b:91044) 18

19 [18] F. V. Fomin, P. A. Golovach, J. Kratochvíl, N. Nisse, an K. Suchan, Pursuing a fast robber on a graph, Theoret. Comput. Sci. 411 (2010), no. 7-9, MR [19] P. Frankl, Cops an robbers in graphs with large girth an Cayley graphs, Discrete Appl. Math. 17 (1987), no. 3, MR (88f:90204) [20] A. Frieze, M. Krivelevich, an P. Loh, Variations on cops an robbers, J. Graph Theory 69 (2012), no. 4, [21] T. Gavenčiak, Catching a fast robber on interval graphs, Theory an applications of moels of computation, Lecture Notes in Comput. Sci., vol. 6648, Springer, Heielberg, 2011, pp MR [22] G. Hahn, Cops, robbers an graphs, Tatra Mt. Math. Publ. 36 (2007), MR (2009b:05254) [23] V. Isler, S. Kannan, an S. Khanna, Ranomize pursuit-evasion with local visibility, SIAM J. Discrete Math. 20 (2006), no. 1, (electronic). MR (2007h:91038) [24] L. Lu an X. Peng, On Meyniel s conjecture of the cop number, J. Graph Theory, to appear. [25] T. Luczak an P. Pra lat, Chasing robbers on ranom graphs: zigzag theorem, Ranom Structures Algorithms 37 (2010), no. 4, MR (2011k:05222) [26] A. Mehrabian, Cops an robber game with a fast robber on expaner graphs an ranom graphs, Annals of Combinatorics, to appear. [27] A. Mehrabian, Cops an robber game with a fast robber, Master s thesis, University of Waterloo, Waterloo, Canaa, March [28] R. Nowakowski an P. Winkler, Vertex-to-vertex pursuit in a graph, Discrete Math. 43 (1983), no. 2-3, MR (84:05138) [29] D. Peleg an J. D. Ullman, An optimal synchronizer for the hypercube, SIAM J. Comput. 18 (1989), no. 4, MR (91a:68214) [30] P. Pra lat, When oes a ranom graph have constant cop number?, Australas. J. Combin. 46 (2010), MR

20 [31] P. Pra lat an N. Wormal, Meyniel s conjecture hols in ranom graphs, preprint. [32] A. Quilliot, Jeux et pointes fixes sur les graphes, Ph.D. thesis, Université e Paris VI, [33] A. Scott an B. Suakov, A boun for the cops an robbers problem, SIAM J. Discrete Math. 25 (2011), no. 3, MR [34] P. D. Seymour an R. Thomas, Graph searching an a min-max theorem for treewith, J. Combin. Theory Ser. B 58 (1993), no. 1, MR (94b:05197) 20