In this short note, we supply a new upper bound on the cop number in terms of
tree decompositions. Our results in some cases extend a previously derived
bound on the cop number using treewidth

Bonato, Anthony

Clarke, N. E.

Finbow, S.

Fitzpatrick, S.

Messinger, M. E.

English

arXiv

In this short note, we supply a new upper bound on the cop number in terms of tree decompositions. Our results in some cases extend a previously derived bound on the cop number using treewidth

Clarke, Nancy E

Finbow, Stephen

Fitzpatrick, Shannon

Messinger, Margaret-Ellen

University of Calgary Journal Hosting

A note on bounds for the cop number using tree decompositions

Contributions to Discrete Mathematics (E-Journal, University of Calgary)

Volume 9, Number 2, Pages 50–56ISSN 1715-0868A NOTE ON BOUNDS FOR THE COP NUMBER USINGTREE DECOMPOSITIONSANTHONY BONATO, NANCY E. CLARKE, STEPHEN FINBOW,SHANNON FITZPATRICK, AND MARGARET-ELLEN MESSINGERAbstract. In this short note, we supply a new upper bound on thecop number in terms of tree decompositions. Our results in some casesextend a previously derived bound on the cop number using treewidth.1. IntroductionThe game of Cops and Robbers (defined at the end of this section) isusually studied in the context of the minimum number of cops needed tohave a winning strategy, or cop number. The cop number (written c(G)for a graph G) is a challenging graph parameter for a variety of reasons,and establishing upper bounds for this parameter are the focus of Meyniel’sconjecture: the cop number of a connected n-vertex graph is O(√n). Foradditional background on both Cops and Robbers and Meyniel’s conjecture,see the recent book [3].The following elegant upper bound was given in [6]:(1) c(G) ≤ tw(G)/2 + 1,where tw(G) is the treewidth of G. The bound (1) is tight if the graph hassmall treewidth (up to treewidth 5). Further, it gives a simple proof thatouterplanar graphs have cop number at most 2 (first proved in [4]).For many families of graphs, however, the bound (1) is far from tight; forexample, for a positive integer n, a clique Kn has treewidth n − 1, but iscop-win. Similarly, Cartesian n × n grids PnPn have cop number 2, buthave treewidth n.In this short note, we give a new bound on the cop number that exploitstree decompositions, and in some cases improves on (1). The idea behindthe proof of (1) is to guard bags and use isometric paths to move cops fromone bag to another. We modify this approach using the notion of a retract,and a retract cover of a graph (see Theorems 1, 2, and 3). Besides givingReceived by the editors November 25, 2012, and in revised form February 14, 2012.2010 Mathematics Subject Classification. 05C57.Key words and phrases. Cops and Robbers, cop number, treewidth, retract, treedecomposition.Supported by grants from NSERC and Ryerson.c©2014 University of Calgary50COP NUMBER AND TREE DECOMPOSITIONS 51the optimal bounds for various families (such as grids, cliques, and k-trees),our results give a new approach to bounding the cop number by exploitingproperties of tree decompositions.1.1. Definitions and notation. We consider only finite, reflexive, undi-rected graphs in the paper. For background on graph theory, the reader isdirected to [5, 9].The game of Cops and Robbers was independently introduced in [7, 8] andthe cop number was introduced in [1]. The game is played on a reflexivegraph; that is, vertices each have at least one loop. Multiple edges are al-lowed, but make no difference to the gameplay, so we always assume there isexactly one edge between adjacent vertices. There are two players consistingof a set of cops and a single robber. The game is played over a sequence ofdiscrete time-steps or rounds, with the cops going first in round 0 and thenplaying alternate time-steps. The cops and robber occupy vertices; for sim-plicity, we often identify the player with the vertex they occupy. We refer tothe set of cops as C and the robber as R. When a player is ready to move ina round they must change vertices to a neighbouring vertex. Because of theloops, players can pass, or remain on their own vertex. Observe that anysubset of C may move in a given round. The cops win if after some finitenumber of rounds, one of them can occupy the same vertex as the robber (ina reflexive graph, this is equivalent to the cop landing on the robber). Thisis called a capture. The robber wins if he can evade capture indefinitely.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 awell-defined positive integer named the cop number (or copnumber) of thegraph G. We write c(G) for the cop number of a graph G. In the specialcase c(G) = 1, we say G is cop-win.An induced subgraph H of G is a retract if there is a homomorphism ffrom G onto H so that f(x) = x for x ∈ V (H); that is, f is the identityon H. The map f is called a retraction. For example, each isometric pathis a retract (as first shown in [1]), as is each clique. Each retract H withretraction f can be guarded by a set of cops in the following sense: if therobber is on x, then the cops play in H as if the robber were on f(x). Ifthere are a sufficient number of cops to capture the image of the robber, then,after finitely many rounds, if the robber entered H he would be immediatelycaught. We denote the minimum number of cops needed to guard H byguard(H). Note this parameter is well-defined, as each vertex of H canbe guarded. In the case of an isometric path P , it was shown in [1] thatguard(P ) = 1.2. Tree decompositionsIn a tree decomposition, each vertex of the graph is represented by asubtree such that vertices are adjacent only when the corresponding subtreesintersect. Formally, given a graph G = (V,E), a tree decomposition is a pair52 A. BONATO, N.E. CLARKE, S. FINBOW, S. FITZPATRICK, AND M.E. MESSINGER(X,T ), where X = {X1, . . . , Xn} is a family of subsets of V called bags, andT is a tree whose vertices are the subsets Xi, satisfying the following threeproperties.(1) Each graph vertex is associated with at least one tree vertex, thatis, V =⋃ni=1Xi.(2) For every edge (v, w) in the graph, there is a subset Xi that containsboth v and w. That is, vertices are adjacent in G only when thecorresponding subtrees have a vertex in common.(3) If Xi, Xj and Xk are nodes, and Xk is on the path from Xi to Xj ,then Xi ∩Xj ⊆ Xk.Item (3) is equivalent to the fact that for each vertex x, the bags contain-ing x form a subtree of T. The width of a tree decomposition is the size ofits largest set Xi minus one. The treewidth of a graph G, written tw(G), isthe minimum width among all possible tree decompositions of G. For moreon treewidth, see [2, 5].Given an induced subgraph H of G, a cover of H in G, written CG(H) isa set of induced subgraphs {Hi : i ∈ I} of G whose union contains H (notethat the subgraphs Hi need not be disjoint). A retract cover of H in G isa cover where each Hi is a retract; we write CR,G(H) to denote a retractcover. See Figure 1 for an example.Figure 1. The subgraph induced by the white and grayvertices form a retract cover of the subgraph H induced bythe white vertices.Define the retract cover cop number of H byrccG(H) = minCR,G(H)∑Hi∈CR,G(H)guard(Hi),where the minimum ranges over all retract covers CR,G(H) of H in G. Forexample, rccG(H) = 1 in the graph in Figure 1, while rccH(H) = 2. Aretract cover which achieves this minimum is called a minimal retract coverof H. Note that if rccG(H)-many cops are available, then after the finitelymany rounds, the cops can be positioned so that if the robber entered H,he would be immediately caught. For each retract in a retract cover of H,the appropriate number of cops guard that retract. This is an essentialobservation used to prove the following theorem. For a bag B, we use thenotation 〈B〉 for the subgraph induced by B.COP NUMBER AND TREE DECOMPOSITIONS 53Theorem 1. If G is a graph, thenc(G) ≤ 2 minT{maxB∈TrccG(〈B〉)},where the minimum ranges over all tree decompositions T of G.For example, if G is the n×n Cartesian grid PnPn, then the following isa tree decomposition of G into isometric paths. Label the vertices as (i, j),where 1 ≤ i, j ≤ n. For 1 ≤ i ≤ n− 1 and 1 ≤ j ≤ n, consider the pathBi,j = {(i, k) : j ≤ k ≤ n} ∪ {(i + 1, k) : 1 ≤ k ≤ j}.See Figure 2 for an illustration of one such path in the case n = 5. As theFigure 2. The path with white vertices is the bag B2,3.Note that vertices are labelled with increasing values movingfrom left-to-right and bottom-to-top (with the bottom leftvertex labelled (1, 1)).retract cover cop number of an isometric path is 1, this tree decompositionand Theorem 1 gives an upper bound of 2 for the cop number of grids (andof course, 2 is the optimal value).Proof. Let m = minT {maxB∈T rccG(〈B〉)}, and let T be a fixed tree decom-position of G realizing the minimum. Arbitrarily place 2m cops in a fixedbag B of T (any bag will do, or the cops can all move to a bag in the centreof the tree to speed up capture). We call one team of m cops X and theother X ′. Match each cop C from X with a unique cop C ′ from X ′, so thatthe cops C and C ′ share each other’s positions. (In particular, at this phaseof the cops’ strategy, there are at least two cops at any position occupied bythe cops). After a finite number of rounds, the cops can position themselveson a minimal retract cover of 〈B〉 so that 〈B〉 is guarded. Hence, the robberR cannot enter B.Let B′ be the unique bag adjacent to B in T which is on a shortestpath connecting the bag B to a bag containing the robber. Note that sincethe set of bags containing the robber is a subtree of T , the robber is notnecessarily in a unique bag. However, as T is a tree, there is a unique bagneighbouring B which has shortest distance to the subtree containing therobber. The cops would like to move to B′ in such a manner that R cannot54 A. BONATO, N.E. CLARKE, S. FINBOW, S. FITZPATRICK, AND M.E. MESSINGERenter B. The cops in team X remain in the minimal retract cover of 〈B〉and continue to guard it; the team X ′ moves to a minimal retract cover of〈B′〉 and after finitely many rounds, guards 〈B′〉. Note that B ∩B′ remainsguarded throughout the transition. The cops in X are now free to movewithout concern that the robber R will enter B.Now the cops in team X can then move to a minimal retract cover CR,G(H)of 〈B′′〉 where B′′ is the unique bag adjacent to B′ which is on a shortestpath in T to a bag containing the robber. Team X then guards CR,G(H).Note that we may now swap the roles of X and X ′, and we have moved thecops closer to the robber in the tree T. We call the process of moving X froma minimal retract cover of B to a minimal retract cover of B′′ a leap step,as the cops in X move through B′ and onwards toward B′′ (see Figure 3).XBX ′B′ B′′Figure 3. A leap stepBy the definition of tree decomposition, the bags containing R form asubtree T ′ of T. By an iterated application of leap steps (after each suchstep, we swap the roles of teams X and X ′) for each bag on a shortestpath connecting the cops’ bag to the robbers, the cops move closer to T ′,ensuring the robber will never enter bags previously guarded by the cops.By induction on the number of vertices of T , the cops may capture therobber. Observe that the proof of Theorem 1 gives an algorithm for capturingthe robber. Further, we can estimate the length of the game using thisalgorithm. To be more precise, the length of a game is the number of roundsit takes (not including the initial, or 0th round) to capture the robber. Wesay that a play of the game with c(G) cops is optimal if its length is theminimum over all possible strategies for the cops, assuming the robber istrying to evade capture for as long as possible. There may be many optimalplays possible (for example, on P4, the cop may start on either vertex of thecentre), but the length of an optimal game is an invariant of G. We denotethis invariant capt(G), which we call the capture time of G. For a bound onthe capture time in terms of the strategy in the proof of Theorem 1, notethat the cops move to a bag in the centre of T , guard that bag, then movetowards the robber’s bag. Given a tree decomposition T , let the numberof rounds it takes to guard a minimal retract cover of any bag in T be atmost gT . Let trT be the number of rounds it takes to move from a minimalCOP NUMBER AND TREE DECOMPOSITIONS 55retract cover of a bag B to a minimal retract cover of a bag which is at mostdistance two from B in T (for instance, as in a leap step). Since the capturetime of T is ddiam(T )/2e and the worst case is that the cops will need toguard each bag and transition along each edge of a path with that length,a bound on the capture time of G is then(2) capt(G) ≤ minT{gT (ddiam(T )/2e+ 1) + trT (ddiam(T )/2e)},where the minimum ranges over all tree decompositions T of G.The bound (2) may be far from tight because it depends on the values ofthe functions gT and trT . We can make a minor improvement on (2) in thecase when diam(T ) is odd (which implies that the centre of T consists of twovertices). Start each of the two teams of cops on a minimal retract cover ofdifferent bags associated with the two bags of the centre of T . After each ofthese bags is guarded, the guards may proceed with leap steps. Using thisalgorithm, a bound on the number of steps to capture the robber is found byreplacing the ceiling functions in (2) with floor functions. Nevertheless, (2)represents the first estimate on the capture time we are aware of applicableto diverse families of graphs such as outerplanar graphs.We note that if each bag is a clique, then the idea of the proof shows astrengthened bound.Theorem 2. If G has a tree decomposition with each bag a clique, then thegraph G is cop-win.Proof. The proof is analogous to Theorem 1, except one cop is needed toguard a given bag. That cop can move to B ∩ B′ without concern that Rwill enter B. Theorem 2 gives an alternative proof that chordal graphs are cop-win, aschordal graphs are precisely those graphs with tree decompositions whereeach bag induces a clique (see [2]). Note also that k-trees are chordal, andhave treewidth k; in particular, the bound (1) is linear in k, while Theorem 2requires only one cop.We finish with a bound in the case where there are conditions on theintersection of bags.Theorem 3. If G is a graph with a tree decomposition T with the propertythat any two bags intersect in a clique, thenc(G) ≤ maxB∈TrccG(〈B〉) + 1.Proof. The proof is analogous to the proof of Theorem 1, except the teamX ′ consists of one additional cop C ′. Using the notation of the proof ofTheorem 1, m-many cops guard 〈B〉 , while the cop C ′ moves to B ∩ B′.Hence, 〈B〉 is guarded. The cops X can move to a minimal retract coverof 〈B′〉 and C ′ ensures that R never enters B (note that B ∩ B′ is a cut-set of G). The cops may now iterate this procedure and, by induction on56 A. BONATO, N.E. CLARKE, S. FINBOW, S. FITZPATRICK, AND M.E. MESSINGERthe number of vertices of the tree decomposition, eventually capture therobber. AcknowledgementsThe authors would like to thank BIRS where part of the research for thispaper was conducted.References[1] M. Aigner and M. Fromme, A game of cops and robbers, Discrete Applied Mathematics8 (1984), 1–12.[2] H.L. Bodlaender, A tourist guide through treewidth, Acta Cybernetica 11 (1993), 1–23.[3] A. Bonato and R.J. Nowakowski, The game of cops and robbers on graphs, StudentMathematical Library, vol. 61, American Mathematical Society, 2011.[4] N.E. Clarke, Constrained cops and robber, Ph.D. dissertation, Dalhousie University,2002.[5] R. Diestel, Graph theory, Graduate Texts in Mathematics, vol. 173, Springer-Verlag,2000.[6] G. Joret, M. Kamin´ski, and D.O. Theis, The cops and robber game on graphs withforbidden (induced) subgraphs, Contributions to Discrete Mathematics 5 (2010), 40–51.[7] R.J. Nowakowski and P. Winkler, Vertex-to-vertex pursuit in a graph, Discrete Math-ematics 43 (1983), 235–239.[8] A. Quilliot, Jeux et pointes fixes sur les graphes, the`se de 3e`me cycle, Ph.D. disserta-tion, Universite´ de Paris VI, 1978.[9] D.B. West, Introduction to graph theory, 2nd ed., Prentice Hall, 2001.Department of Mathematics, Ryerson University, Toronto, ON, Canada,M5B 2K3E-mail address: abonato@ryerson.caDepartment of Mathematics & Statistics, Acadia University, Wolfville, NS,Canada, B4P 2R6E-mail address: nancy.clarke@acadiau.caDepartment of Mathematics, Statistics, & Computer Science, St. FrancisXavier University, Antigonish, NS, Canada, B2G 2W5E-mail address: sfinbow@stfx.caDepartment of Mathematics & Statistics, University of Prince EdwardIsland, Charlottetown, PE, Canada, C1A 4P3E-mail address: sfitzpatrick@upei.caDepartment of Mathematics & Computer Science, Mount Allison University,Sackville, NB, Canada, E4L 1E6E-mail address: mmessinger@mta.ca

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A note on bounds for the cop number using tree decompositions

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