The robber strikes back

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1 The robber strkes back Anthony Bonato 1, Stephen Fnbow 2, Przemysław Gordnowcz 3, Al Hadar 4, Wllam B. Knnersley 1, Deter Mtsche 5, Paweł Prałat 1, and Ladslav Stacho 6 1 Ryerson Unversty 2 St. Francs Xaver Unversty 3 Techncal Unversty of Lodz 4 Carleton Unversty 5 Unversty of Nce Sopha-Antpols 6 Smon Fraser Unversty Abstract We consder the new game of Cops and Attackng Robbers, whch s dentcal to the usual Cops and Robbers game except that f the robber moves to a vertex contanng a sngle cop, then that cop s removed from the game. We study the mnmum number of cops needed to capture a robber on a graph G, wrtten cc(g). We gve bounds on cc(g) n terms of the cop number of G n the classes of bpartte graphs and dameter two, K 1,m-free graphs. AMS 2010 Subject Classfcaton: 05C57 Keywords: Cops and Robbers, cop number, bpartte graphs, claw-free graphs. 1 Introducton Cops and Robbers s a vertex-pursut game played on graphs that has been the focus of much recent attenton. Throughout, we only consder fnte, connected, and smple undrected graphs. There are two players consstng of a set of cops and a sngle robber. The game s played over a sequence of dscrete tme-steps or rounds, wth the cops gong frst n the frst round and then playng on alternate tme-steps. The cops and robber occupy vertces, and more than one cop may occupy a vertex. When a player s ready to move n a round they may move to a neghbourng vertex or pass by remanng on ther own vertex. Observe that any subset of cops may move n a gven round. The cops wn f after some fnte number of rounds, one of them can occupy the same vertex as the robber. Ths s called a capture. The robber wns f he can avod capture ndefntely. A wnnng strategy for the cops s a set of rules that f followed result n a wn for the cops, and a wnnng strategy for the robber s defned analogously. If we place a cop at each vertex, then the cops are guaranteed to wn. Therefore, the mnmum number of cops requred to wn n a graph G s a well defned postve nteger, named the cop number of the graph G. We wrte c(g) for the cop number of a graph G. For example, the Petersen graph has cop number 3. Nowakowsk and Wnkler [14], and ndependently Qullot [19], consdered the game wth one cop only; the ntroducton of the cop number came n [1]. Many papers have now been wrtten on cop number snce these three early works; see the book [8] for addtonal references and background on the cop number. See also the surveys [2, 4, 5]. The authors were supported by grants from NSERC and Ryerson Unversty. 1

2 Many varants of Cops and Robbers have been studed. For example, we may allow a cop to capture the robber from a dstance k, wherek s a non-negatve nteger [7], play on edges [12], allow one or both players to move wth dfferent speeds or teleport, or allow the robber to be nvsble. See Chapter 8 of [8] for a non-comprehensve survey of varants of Cops and Robber. We consder a new varant of the game of Cops and Robbers, where the robber s able to essentally strke back aganst the cops. We say that the robber attacks acopfhechoosestomovetoavertexonwhcha cop s present and elmnates her from the game. In the game of Cops and Attackng Robbers, the robber may attack a cop, but cannot start the game by movng to a vertex occuped by a cop; all other rules of the game are the same as n the classc Cops and Robbers. We note that f two cops are on a vertex u and the robber moves to u, then only one cop on u s elmnated; the remanng cop then captures the robber and the game ends. We wrte cc(g) for the mnmum number of cops needed to capture the robber. Note that cc(g) s the analogue of the cop number n the game of Cops and Attackng Robber; our choce of notaton wll be made more transparent once we state Theorem 1. We refer to cc(g) as the cc-number of G. Snce placng a cop on each vertex of G results n a wn for the cops, the parameter cc(g) s well-defned. To llustrate that cc(g) can take dfferent values from the cop number, consder that for the cycle C n wth n vertces we have the followng equaltes (whch are easly verfed): 1 f n =3, cc(c n )= 2 f 4 n 6, 3 else. We outlne some basc results and bounds for the cc-number n Secton 2. We consder bounds on cc(g) n terms of c(g) n Secton 3. In Secton 4 we gve the bound of cc(g) c(g) +2 nthecasethatg s bpartte; see Theorem 9. In the fnal secton, we supply n Theorem 10 an upper bound for cc(g) for K 1,m -free, dameter two graphs. For background on graph theory see [20]. For a vertex u, weletn(u) denote the neghbour set of u, and N[u] =N(u) {u} denote the closed neghbour set of u. The set of vertces of dstance 2 to u s denoted by N 2 (u). We denote by δ(g) the mnmum degree n G. In a graph G, asets of vertces s a domnatng set f every vertex not n S has a neghbor n S. The domnaton number of G, wrtten γ(g), s the mnmum cardnalty of a domnatng set. The grth of a graph s the length of the shortest cycle contaned n that graph, and s f the graph contans no cycles. 2 Basc results In ths secton we collect together some basc results for the cc-number. As the proofs are ether elementary or mnor varatons of the analogous proofs for the cop number, they are omtted. The frst result on the game of Cops and Attackng Robbers s the followng theorem; note that the second nequalty naturally nspres the notaton cc(g). We use the notaton c(g) for the edge cop number, whch s a varant where the cops and robber move on edges; see [12]. Theorem 1. If G s a graph, then c(g) cc(g) mn{2c(g), 2 c(g),γ(g)}. The followng theorem s foundatonal n the theory of the cop number. Theorem 2. [1] If G has grth at least 5, then c(g) δ(g). The followng theorem extends ths result to the cc-number. 2

3 Theorem 3. If G has grth at least 5, then cc(g) δ(g)+1. Isometrc paths play an mportant role n several key theorems n the game of Cops and Robbers, such as the cop number of planar graphs (see Chapter 4 of [8]). We call a path P n a graph G sometrc f the shortest dstance between any two vertces s equal n the graph nduced by P and n G. For a fxed nteger k 1, an nduced subgraph H of G s k-guardable f, after fntely many moves, k cops can move only n the vertces of H n such a way that f the robber moves nto H at round t, then he wll be captured at round t +1by a cop n H. For example, a clque n a graph s 1-guardable. Agner and Fromme [1] proved the followng result. Theorem 4. [1] An sometrc path s 1-guardable. We have an analogue of Theorem 4 for the cc-number. Theorem 5. An sometrc path s 2-guardable n the game of Cops and Attackng Robbers, but need not be 1-guardable. See Fgure 1 for an example where the robber can freely move onto an sometrc path wthout beng captured by a sole cop. R C Fgure 1: One cop cannot guard the sometrc path (depcted n bold). We assume that the robber has just arrved at ther vertex and t s the cop s turn to move. AgraphG s called planar f t can be embedded n a plane wthout two of ts edges crossng. It was shown frst n [1] that planar graphs requre at most three cops to catch the robber; see [8] for an alternatve proof of ths fact. Gven the results above, we may conjecture that the cc-number of a planar graph s at most 4 or even 5, but ether bound remans unproven. Outerplanar graphs are those that can be embedded n the plane wthout crossngs n such a way that all of the vertces belong to the unbounded face of the embeddng. Clarke proved the followng theorem n her doctoral thess. Theorem 6. [11] If G s outerplanar, then c(g) 2. The counterpart to Theorem 6 s the followng. Theorem 7. If G s outerplanar, then cc(g) 3. Meynel s conjecture frst communcated by Frankl [13] s one of the most mportant open problems surroundng the game of Cops and Robbers. The conjecture states that c(n) =O( n), wherec(n) s the maxmum of c(g) over all n-vertex, connected graphs. Cops and Robbers has been studed extensvely for random graphs (see for example, [3, 9, 15, 16]), partly owng to a search for counterexamples to Meynel s conjecture. However, t was recently shown that Meynel s conjecture holds asymptotcally almost surely (that s, wth probablty tendng to 1 as the number of vertces tends to nfnty) for both bnomal random graphs G(n, p) [17] as well as random d-regular graphs [18]. 3

4 In [9] t was shown that for dense random graphs, where p = n o(1) and p<1 ɛ for some ɛ>0, asymptotcally almost surely we have that c(g(n, p)) = (1 + o(1))γ(g(n, p)) = (1 + o(1)) log 1/(1 p) n, (1) Note that (1) mples that c(g(n, p)) = (1 + o(1))cc(g(n, p)) for the stated range of p; n partcular, applyng (1) to the p =1/2 case (whch corresponds to the unform probablty space of all labelled graphs on n vertces), we have that for every ɛ>0, almost all graphs satsfy cc(g)/c(g) [1, 1+ɛ]. Unfortunately, the asymptotc value of the cop number s not known for sparser graphs. However, t may be provable that c(g(n, p)) = (1 + o(1))cc(g(n, p)) for sparse graphs, wthout fndng an asymptotc value. We fnsh the secton by notng that graphs wth cc(g) =1are precsely those wth a unversal vertex. However, characterzng those graphs G wth cc(g) =2s an open problem. Graphs wth cc(g) =2nclude cop-wn graphs wthout unversal vertces, and graphs whch are not cop-wn but have domnaton number 2. Before the reader conjectures ths gves a characterzaton, note that the graph n Fgure 2 wth cc-number equalng 2 s n nether class. Fgure 2: A graph G wth c(g) =cc(g) =2and γ(g) =3. 3 How large can the cc-number be? One of the man unanswered questons on the game of Cops and Attackng Robbers s how large the ccnumber can be relatve to the cop number. Many of the results from the last secton mght lead one to (mstakenly) conjecture that cc(g) c(g)+1 for all graphs, and ths was the thnkng of the authors and others for some tme. We provde a counterexample below. By Theorem 1, we know that cc(g) s bounded above by 2c(G). For example, ths s a tght bound for a path of length at least 3. However, we do not know an mproved bound whch apples to general graphs, nor do we possess graphs G wth c(g) > 2 whose cc-number equals 2c(G). In ths secton, we outlne one approach whch may ultmately yeld such examples. Improved bounds for several graph classes are outlned n the next two sectons. Our constructon utlzes lne graphs of hypergraphs. For a postve nteger k, ak-unform hypergraph has every hyperedge of cardnalty k. A hypergraph s lnear f any two hyperedges ntersect n at most one vertex. The lne graph of a hypergraph H, wrtten L(H), has one vertex for each hyperedge of H, wth two vertces adjacent f the correspondng hyperedges ntersect. Lemma 8. Let H be a lnear k-unform hypergraph wth mnmum degree at least 3 and grth at least 5. If L(H) has domnaton number at least 2k, then cc(l(h)) 2k. 4

5 Proof. Suppose there are at most 2k 1 cops. Snce the domnaton number of L(H) s at least 2k, the robber can choose an ntal poston that lets hm survve the cops frst move. To show that 2k 1 cops cannot catch the robber n the game of Cops and Attackng Robber on L(H), suppose otherwse, and consder the state of the game on the robber s fnal turn (that s, just before he s to be captured). Let v be the robber s current vertex, E v the correspondng edge of H, andw 1,w 2,...,w k the elements of E v. The neghbours of v n L(H) are precsely those vertces correspondng to edges of H that ntersect E v ;denotebys w the set of vertces (other than v) correspondng to edges contanng w. Each S w s a clque; moreover, snce H has mnmum degree at least 3, each contans at least two vertces. By hypotheses for H, t follows that the S w are dsjont and that no vertex outsde S w domnates more than one vertex nsde. Fnally, snce H has grth at least 5, no vertex n G domnates vertces n two dfferent S w (that s, the neghbourhoods N[S w ] only have v n common). Consder the cops current postons. The cops must domnate all of N[v], snce otherwse the robber would be able to survve for one more round (by movng to an undomnated vertex). Snce the N[S w ] only have v n common, for some j we have at most one cop n N[S wj ]. If n fact there are no cops n N[S wj ], then no vertces of S wj are domnated, a contradcton. Thus, S wj contans exactly one cop. Snce each vertex outsde S wj domnates at most one vertex nsde and S wj contans at least two vertces, the cop must actually stand wthn S wj. However, snce she s the only cop wthn N[S wj ], the robber may attack the cop wthout leavng hmself open to capture on the next turn. Thus, the robber always has a means to avod capture on the cops next turn. Hence, at least 2k cops are needed to capture the robber, as clamed. We am to fnd, for all k, graphs G such that c(g) =k and cc(g) =2k. Ths, however, remans open for all k 3. As an applcaton of the lemma, take H to be the Petersen graph. It s easly verfed that c(l(h)) = 2; see also [12]. Lemma 8 wth k =2shows that cc(l(h)) 4; hence, Theorem 1 then mples that cc(l(h)) = 4. See Fgure 3 for a drawng of the lne graph of the Petersen graph. Fgure 3: The lne graph of the Petersen graph. 4 Bpartte graphs For bpartte graphs, we derve the followng upper bound. Theorem 9. For every connected bpartte graph G, we have that cc(g) c(g)+2. Proof. Fx a connected bpartte graph G. Letk = c(g); wegveastrategyfork +2 cops to wn the game of Cops and Attackng Robbers on G. Label the cops C 1,C 2,...,C k,c 1,C 2. Intutvely, cops C 1,C 2,...,C k attempt to follow a wnnng strategy for the ordnary Cops and Robber game on G; snce they must avod beng klled by the robber, they may not be able to follow ths strategy exactly, but can follow t closely enough. Cops C 1 and C 2 play a dfferent role: they occupy a common vertex throughout the game, and n each round, they smply move closer to the robber. Ths has the effect of eventually forcng the robber to 5

6 move on every turn. (Snce the cops move together, the robber cannot safely attack ether one.) Further, when the robber passes, the cops C 1,C 2,...,C k,c1,c2 pass. Therefore, we may suppose throughout that the robber moves to a new vertex on each turn. It remans to formally specfy the movements of C 1,C 2,...,C k. To each cop C, we assocate a shadow S. Throughout the game the shadows follow a wnnng strategy for the ordnary game on G. LetC (t), S (t), and R (t) denote the postons of C, S, and the robber, respectvely, at the end of round t. We mantan the followng nvarants for 1 k and all t: 1. S (t) 2. f C (t+1) 3. C (t+1) N[C (t) ] (that s, each cop remans on or adjacent to her shadow); S (t+1),thens (t+1) and R (t) belong to dfferent partte sets of G; s not adjacent to R (t) (that s, the robber never has the opportunty to attack any cop). On round t +1, each cop C moves as follows: (a) If C (t) (b) f C (t) S (t),thenc moves to S (t) ; = S (t),ands (t+1) s not adjacent to R (t),thenc moves to S (t+1) ; (c) otherwse, C remans at her current vertex. By nvarant (1), ths s clearly a legal strategy. We clam that all three nvarants are mantaned. Invarant (1) s straghtforward to verfy. For nvarant (2), frst suppose that C (t) = S (t), but C (t+1) S (t+1). By the cops strategy, ths can happen only when S (t+1) s adjacent to R (t), n whch case the shadow and robber belong to dfferent partte sets, as desred. Now suppose that C (t) S (t) and C (t+1) S (t+1). By the cops strategy we have C (t+1) = S (t). It follows that C (t+1) C (t), S (t+1) S (t),andr (t 1) R (t).thus,fs (t) and R (t 1) belong to dfferent partte sets, then so must S (t+1) and R (t) ; that s, the nvarant s mantaned. For nvarant (3), f S (t+1) s adjacent to R (t), then we may suppose that S (t+1) S (t), snce otherwse the shadow would have captured the robber n round t +1. By the cops strategy, we now have that C (t+1) S (t+1). But now the cop and her shadow are n dfferent partte sets by nvarant (1), and the shadow and robber are n dfferent partte sets by nvarant (2), so the cop and robber are n the same partte set, contradctng adjacency of the cop and the robber. Snce the shadows follow a wnnng strategy, eventually some shadow S captures the robber; that s, for some t, we have that ether S (t) = R (t) or S (t+1) = R (t). In the former case, nvarant (3) mples that C (t) S (t) and nvarant (1) mples that C captures the robber n round t +1. Now consder that case when S (t+1) = R (t). By nvarant (2), snce S (t+1) s not adjacent to R (t),wenfacthavethat C (t+1) = S (t+1) = R (t), so the cops have won. 5 K 1,m -free, dameter 2 graphs We provde one more result gvng an upper bound on the cc-number for a set of graph classes. Theorem 10. Let G be a K 1,m -free, dameter 2 graph, where m 3. Then cc(g) c(g)+2m 2. When m =3, Theorem 10 apples to claw-free graphs; see [10] for a characterzaton of these graphs. The cop number of dameter 2 graphs was studed n [6]. 6

7 Proof of Theorem 10. AcopC s back-up toacopc f C s n N[C ]. Note that a cop wth a back-up cannot be attacked wthout the robber beng captured n the next round. Now let c(g) =r, and consder c(g) cops labelled C 1,C 2,...,C r. We refer to these r-many cops as squad 1. Label an addtonal 2m 2 cops as Ĉ,1 and Ĉ,2, where 1 m 1; these cops form squad 2. The ntuton behnd the proof s that the cops n squad 2 act as back-up for those n squad 1, who play ther usual strategy on G. Further, the cops Ĉ,j are postoned n such a way that the cops C k need only restrct ther movements to the second neghbourhood of some fxed vertex. More explctly, fx a vertex x of G. Move squad 2 so that they are contaned n N[x]. Next, poston each of the cops Ĉ,1 on x. Hence, R must reman n N 2 (x) or he wll lose n the next round (n partcular, no squad 2 cop s ever attacked). Throughout the game we wll always mantan the property that there are m 1 cops on x. We note that the squad 2 cops n N(x) can move there essentally as f that subgraph were a clque, and n addton, preserve the property that m 1 cops reman on x. To see ths, f Ĉ,2 were on y N(x) and the cops would lke to move to z N(x), then move Ĉ,2 to x, and move some squad 2 cop from x to z. In partcular, a cop from squad 2 can arrange thngs so that she s adjacent to a cop n squad 1 after at most one move. We refer to ths movement of the squad two cops as a hop, as the cops appear to jump from one vertex of N(x) to another (although what s really happenng s that the cops are cyclng through x). Note that hops mantan m 1 cops on x. We now descrbe a strategy S for the cops, and then show that t s wnnng. The cops n squad 1 play exactly as n the usual game of Cops and Robbers; note that the squad 1 cops may leave N 2 (x) dependng on ther strategy, but R wll never leave N 2 (x). The squad 2 cops play as follows. Squad 2 cops do not move unless the followng occurs: a squad 1 cop C k moves to a neghbour of R, and C k has no back-up from a squad 1 cop. In that case, some squad 2 cop Ĉ,j hops to a vertex of N(x) whch s adjacent to C k. There are a suffcent number of squad 2 cops to ensure ths property, snce f m (or more) squad 1copsmoveto neghbours of R, then some of these cops must be adjacent to each other as G s K 1,m -free (n partcular, the cops n N(R) play the role of back-ups to each other). Hence, the squad 1 cops may apply ther wnnng strategy n the usual game and ensure that whenever they move to a neghbour of R, some squad 2 cop serves as back-up. In partcular, R wll never attack a squad 1 cop for the duraton of the game. Thus, S s a wnnng strategy n the game of Cops and Attackng Robbers. References [1] M. Agner, M. Fromme, A game of cops and robbers, Dscrete Appled Mathematcs 8 (1984) [2] W. Bard, A. Bonato, Meynel s conjecture on the cop number: a survey, Journal of Combnatorcs 3 (2012) [3] B. Bollobás, G. Kun, I. Leader, Cops and robbers n a random graph, Journal of Combnatoral Theory Seres B 103 (2013) [4] A. Bonato, WHAT IS... Cop Number? Notces of the Amercan Mathematcal Socety 59 (2012) [5] A. Bonato, Catch me f you can: Cops and Robbers on graphs, In: Proceedngs of the 6th Internatonal Conference on Mathematcal and Computatonal Models (ICMCM 11), [6] A. Bonato, A. Burgess, Cops and Robbers on graphs based on desgns, Journal of Combnatoral Desgns 21 (2013) [7] A. Bonato, E. Chnforooshan, P. Prałat, Cops and Robbers from a dstance, Theoretcal Computer Scence 411 (2010)

8 [8] A. Bonato, R.J. Nowakowsk, The Game of Cops and Robbers on Graphs, Amercan Mathematcal Socety, Provdence, Rhode Island, [9] A. Bonato, P. Prałat, C. Wang, Network securty n models of complex networks, Internet Mathematcs 4 (2009) [10] M. Chudnovsky, P. Seymour, Clawfree Graphs IV - Decomposton theorem, Journal of Combnatoral Theory. Ser B 98 (2008) [11] N.E. Clarke, Constraned Cops and Robber, Ph.D. Thess, Dalhouse Unversty, [12] A. Dudek, P. Gordnowcz, P. Prałat, Cops and Robbers playng on edges, Preprnt [13] P. Frankl, Cops and robbers n graphs wth large grth and Cayley graphs, Dscrete Appled Mathematcs 17 (1987) [14] R.J. Nowakowsk, P. Wnkler, Vertex-to-vertex pursut n a graph, Dscrete Mathematcs 43 (1983) [15] T. Łuczak, P. Prałat, Chasng robbers on random graphs: zgzag theorem, Random Structures and Algorthms 37 (2010) [16] P. Prałat, When does a random graph have constant cop number?, Australasan Journal of Combnatorcs 46 (2010) [17] P. Prałat, N.C. Wormald, Meynel s conjecture holds for random graphs, Preprnt, [18] P. Prałat, N.C. Wormald, Meynel s conjecture holds for random d-regular graphs, Preprnt [19] A. Qullot, Jeux et pontes fxes sur les graphes, Thèse de 3ème cycle, Unversté de Pars VI, 1978, [20] D.B. West, Introducton to Graph Theory, 2nd edton, Prentce Hall,

COPS AND ROBBER WITH CONSTRAINTS

COPS AND ROBBER WITH CONSTRAINTS FEDOR V. FOMIN, PETR A. GOLOVACH, AND PAWE L PRA LAT Abstract. Cops & Robber s a classcal pursut-evason game on undrected graphs, where the task s to dentfy the mnmum number