arXiv:2001.00477v1 [math.CO] 30 Dec 2019 obrwn fh a vi en atrd h o ubrof number cop The o captured. one time, being in avoid sta point can he any he turn, at if if robber’s wins cop win the robber of cops In turn The the . th vertex. In neighboring then neighboring a and cop. to start, the moves to with or vertex starting a turns chooses alternate cops they the of Each robber. ubro osnee ota h oshv inn strategy winning a have cops the that so needed cops of number H aeo osadrbes Aqikpie ntecpnme s[ is number fanta cop a the is on [3] primer Nowakowski quick and (A Bonato Som robbers. by and understood. book cops clearly The of not game is [9]. number and cop [1] high in have to graph a of P ugah obde.Jrt aisi n hi 7 proved [7] Theis and t Kaminski, fo that Joret, strategy show winning to forbidden. simple is subgraphs conceptually note a this give of to purpose exploited The become [10]). has Gy´arf´as (see argument The coloring path [10]). (see argument rp ihu oe flnt tleast at length of holes without graph a u h hoe oiial ojcueo Gy´arf´as) of that conjecture a “ (originally is theorem the but for etxsti rp ssi ob once ftesbrp i subgraph the if connected be to said pro Gy´arf´as is to by graph [6] used a first was in Gy´arf´as argument set path vertex A o edr aiirwt h yafa ahagmn,cnb rob can Gy´arf´as the argument, the with path t with simple, strengthe familiar conceptually a readers being prove for We of advantage given. are the details has no but works missing) − t fe if -free v ildnt h ahgahon graph path the denote will P h aeo osadrbesi lydo nt ipegraph simple finite a on played is robbers and cops of game The l rpsi hsatceaefiie ipe n connected. and simple, finite, are article this in graphs All χ oso yafa ah n ld h osutlterobber the until cops the slide Gy´arf´as and a path, on cops 3 eoe by denoted , t budd a nyrcnl rvdadnee oe da in ideas novel needed and proved recently only was -bounded” fe rps(hs ro rcesi onsadue induct uses and rounds in proceeds proof (whose graphs -free eas iea nqaiyrltn h o ubradteDlot numb Dilworth the and number cop the relating inequality an give also we o oti oeo egha least at length that of proved hole a contain not for oei rp sa nue yl flnt tlat4 egv ipewin simple a give We 4. least at length of cycle induced an is graph a in hole A t G − osntcontain not does ost atr obri h aeo osadrbespae in played robbers and cops of game the in robber a capture to cops 3 t N − o ubro rpswtotln holes long without graphs of number Cop [ v oshv inn taeyi uhgah.A osqec of consequence a As graphs. such in strategy winning a have cops 2 ,i h e falnihosof neighbors all of set the is ], H sa nue ugah o vertex a For subgraph. induced an as t etcs oein hole A vertices. t hssrntesatermo oe-aisiTes who Joret-Kaminski-Theis, of theorem a strengthens This . ad Sivaraman Vaidy t aur ,2020 3, January hycamta nagmn iia oteterproof their the to similar argument an that claim They . Abstract 1 v including , G h osi rp ihcraninduced certain with graph a in cops the r sa nue yl flnt tlat4. least at length of cycle induced an is ethat ve h ls fgah ihu ogholes long without graphs of class the umrzdcmatya Paethe “Place as compactly summarized e esrcueo h Gy´arf´as be the can of path structure he that ,ec o ihrsaso h vertex the on stays either cop each s, igo hi hoe.Orpofalso proof Our theorem. their of ning e en atrdvr ucl,and quickly, very captured being ber so h aevre rmvst a to moves or vertex same the on ys tnadpoftcnqei graph in technique proof standard a obrcossavre.Adthen And vertex. a chooses robber e h oslnso h obr The robber. the on lands cops the f o graphs For in udmna eut eeproved were results fundamental e t scornered”. is G o,adsm seta eal are details essential some and ion, v 2].) tcsuc fifraino the on information of source stic − dcdo ti once.The connected. is it on nduced G denoted , P G tef o oiieinteger positive a For itself. hr are There . oscncpuearbe in robber a capture can cops 2 t h usino htmakes what of question The . diint h Gy´arf´as the path to addition fe rpsae“ are graphs -free v h lsdneighborhood closed the , ro graph. a of er ,G H, rp htdoes that graph a c ( G k ,i h minimum the is ), igstrategy ning esythat say we osadasingle a and cops u bound, our χ -bounded” G is t , Theorem 0.1. Let G be a graph without holes of length at least t (t ≥ 4). Then t − 3 cops can capture the robber.

Proof. Let v0 ∈ V (G). In the first move player C (the cop player) places all the t − 3 cops in v0. One of the cops is going to be stationary at v0 and the other t − 4 cops will move in the next turn. The robber will choose a vertex in V (G) \ N[v0]. Let v1 be a neighbor of v0 that has a neighbor in the component C1 of G − N[v0] containing the robber vertex. Now player C moves t − 4 of his cops from v0 to v1. Now the robber moves to some vertex (or stays put), and let v2 be a neighbor of v1 that has a neighbor in the component C2 of G[C1] − N[v1] containing the robber vertex. Player C moves t − 5 of his cops from v1 to v2. This procedure is repeated so that at the end of t − 2 moves, we have an induced path v0 − v1 −···− vt−4, a nested sequence of connected vertex sets C1 ⊇ C2 ···⊇ Ct−4, where vi 6∈ Ci but vi has a neighbor in Ci. There is a cop on each of the vertices v0, · · · , vt−4 and the robber is in some vertex in Ct−4. This finishes the first phase.

Now we start the second phase. In every turn, the Gy´arf´as path constructed in the first phase is going to be extended, and every cop is going to move one position to the right along the Gy´arf´as path that is being built. So after one move the cops will be in v1, · · · , vt−3, the robber will be in some vertex in Ct−3, and after the second move (in this phase) the cops will be in v2, · · · , vt−2, the robber will be in some vertex in Ct−2 and so on. After k moves in the second phase, the cops are in vk, · · · , vk+t−4, and the robber is in some vertex in Ck+t−4. The crucial point is that, as we are going to show, the robber cannot escape from Ck+t−4, meaning he cannot move to a vertex not in ′ ′ Ck+t−4. Suppose he does. Say he moves from r to a vertex r 6∈ Ck+t−4. Clearly r is non-adjacent to each of vk, · · · , vk+t−4, for, otherwise the robber will be caught in the next move. By virtue of the special structure of the Gy´arf´as path and its relation to the rest of the graph, r′ is adjacent to some vertex on the Gy´arf´as path v0 −···− vk+t−4. (This is the crucial step that makes the proof work, and explains why the sets Ci are defined in the way mentioned.) Let i be the largest index ′ ′ with i < k such that r is adjacent to vi. Now vi − vi+1 −···− vk+t−4 − P − r − r − vi is a hole of length at least t, a contradiction, where P is a shortest path from vk+t−4 to r with internal vertices in Ck+t−4. Hence after every move (of the cops and the robber), the length of the Gy´arf´as path being built increases. (The robber is forced to do this as he cannot go out of the component he is currently in.) But the length of a Gy´arf´as path in G cannot be more than |V (G)|. The robber will be caught in at most |V (G)| moves. This completes the proof.

The above argument gives a very transparent reason as to why a single cop has a winning strategy in a (the case t = 4). (Recall that a graph is chordal if it has no holes.) The proof presented here is similar to the one in [11] but has a second part where the cops move. This is necessary since the length of an induced path in a graph without a hole of length at least t need not be bounded by a function of t. The path constructed in the proof is called a Gy´arf´as path, but in , the component is chosen to be the one with the largest chromatic number rather than the component containing the robber. Analysis of the above proof inspires one to define the following. For each vertex v, let z(v) denote the length of a longest induced path starting at v. And define z(G) to be the minimum, taken over v ∈ V (G), of z(v). By choosing v0 to be the vertex with minimum z value in the proof above, we see that if G is a graph without holes of length at least t (t ≥ 4) t − 3 cops can capture the robber in at most z(G) moves. This has the following simple corollary. (The bound on the number of cops was proved in [7], and the bound on the number of

2 moves was proved in [11].)

Corollary 0.1. Let G be a Pt-free graph. Then t − 2 cops can capture the robber in at most t − 1 moves.

Proof. Follows from the theorem and the obvious fact that since G is Pt-free, it does not contain a hole of length at least t +1 and z(G) ≤ t − 1.

The simplicity of the argument presented above immediately suggests: Can we do better if we understand more about the structure of graphs not containing a hole of length at least t? Here is a conjecture. Conjecture 0.1. Let G be a graph without a hole of length at least t (t ≥ 6). Then t − 4 cops can capture the robber. The first case, namely that 2 cops can capture a robber in a graph without holes of length at least 6 looks particularly interesting. Conjecture 0.2. Let G be a graph without holes of length at least 6. Then 2 cops can capture the robber. Two weaker conjectures are still open.

Conjecture 0.3. [11] Let G be a P5-free graph. Then 2 cops can capture the robber.

Conjecture 0.4. [12] Let G bea 2K2-free graph. Then 2 cops can capture the robber. It is tempting to ask whether graphs without holes of odd length have bounded cop number. Unfortunately, even triangle-free members of the class, namely bipartite graphs, have unbounded cop number (see [3]). It is equally tempting (by analogy with recent developments in “χ-boundedness”, see [10]) to ask what happens when we exclude a as induced subgraph. Does it force the cop number to be bounded? Well, it is well known that line graphs (which are claw-free) have unbounded cop number (see [3]). This immediately leads to a (possibly very hard) problem. Problem 0.1. Characterize all sets F of graphs such that the class of F-free graphs has bounded cop number. Finally, we derive an inequality relating the cop number and the Dilworth number of a graph. First we need to give a formal definition of the Dilworth number, first defined in [5].

For a vertex v, N(v) will denote the set of neighbors of v and N[v] := {v}∪ N(v). A set U of vertices in a graph G is said to be Dilworth if for any distinct u, v ∈ U, N(u) \ N[v] 6= ∅ and N(v) \ N[u] 6= ∅. The Dilworth number of a graph G, denoted D(G), is the size of a largest Dilworth set. Graphs with Dilworth number 1 are precisely the threshold graphs [4]. A full chapter is devoted to the Dilworth number in [8], a comprehensive treatise on threshold graphs. Theorem 0.2. For a graph G with D(G) ≥ 3, c(G) ≤ D(G) − 2.

Proof. Note that, for k ≥ 4, the Dilworth number of the Ck is k. Also, the Dilworth number is monotone under taking induced subgraphs i.e., the Dilworth number of an induced sub- graph is at most the Dilworth number of the parent graph. Hence a graph G cannot contain a hole of length D(G) + 1. The desired inequality follows immediately from theorem 0.1.

3 It is very unlikely that the above inequality is tight, especially for larger values of D(G). (It is tight for C4.) It might be interesting to characterize graphs G with c(G)= D(G) − 2.

Acknowledgements. The proof in this article was presented at the Barbados Work- shop 2019 in Bellairs Research Institute in Holetown, Barbados, and the feedback from some of the participants was very useful in the preparation of this article. In particular, I would like to thank Paul Seymour, Sophie Spirkl, and Cemil Dibek.

References

[1] M. Aigner, M. Fromme, A game of cops and robbers, Discrete Appl. Math. 8 (1984) 1-11.

[2] A. Bonato, What is ... cop number?, Notices Amer. Math. Soc. 59 no. 8, (2012), 1100-1101.

[3] A. Bonato, R. J. Nowakowski, The game of cops and robbers on graphs, American Mathematical Society, 2011.

[4] V. Chv´atal and P. L. Hammer, Set-packing and threshold graphs, Univ. Waterloo Res. Report CORR (1973), 73–21.

[5] S. Foldes, P. L. Hammer, The Dilworth number of a graph, in Algorithmic aspects of combinatorics, Annals of Discrete Mathematics, vol. 2, 1978, 211-219.

[6] A. Gy´arf´as, Problems from the world surrounding perfect graphs, Zastosowania Matematyki Applica- tionies Mathematicae XIX, 3-4 (1987), 413-441.

[7] G. Joret, M. Kaminski, D. O. Theis, The cops and robber game on graphs with forbidden (induced) subgraphs, Contributions to discrete mathematics 5, Number 2, (2010) 40-51.

[8] N. V. R. Mahadev, U. N. Peled, Threshold graphs and related topics, Annals of Discrete Mathematics, 56, North-Holland Publishing Co., Amsterdam, 1995.

[9] R. Nowakowski, P. Winkler, Vertex-to-vertex pursuit in a graph, Discrete Math. 43 (1983), 235-239.

[10] A. Scott, P. Seymour, A survey of χ-boundedness, submitted. (available at https://web.math.princeton.edu/ pds/papers/chibounded/paper.pdf)

[11] V. Sivaraman, An application of the Gy´arf´as path argument, Discrete Math. 342 (2019) 2306-2307.

[12] V. Sivaraman, S. Testa, Cop number of 2K2-free graphs, submitted.

4