Chasing robbers on random graphs: Zigzag theorem

In this paper, we study the vertex pursuit game of Cops and Robbers where cops try to capture a robber on the vertices of the graph. The minimum number of cops required to win on a given graph G is the cop number of G. We present asymptotic results for the game of Cops and Robber played on a random graph G(n,p) for a wide range of p = p(n). It has been shown that the cop number as a function of an average degree forms an intriguing zigzag shape. © 2010 Wiley Periodicals, Inc. Random Struct. Alg., 2010     

Cops and Robbers on Planar‐Directed Graphs

Aigner and Fromme initiated the systematic study of the cop number of a graph by proving the elegant and sharp result that in every connected planar graph, three cops are sufficient to win a natural pursuit game against a single robber. This game, introduced by Nowakowski and Winkler, is commonly known as Cops and Robbers in the combinatorial literature. We extend this study to directed planar graphs, and establish separation from the undirected setting. We exhibit a geometric construction that shows that a sophisticated robber strategy can indefinitely evade three cops on a particular strongly connected planar‐directed graph.     

The capture time of a graph

We consider the game of Cops and Robbers played on finite and countably infinite connected graphs. The length of games is considered on cop-win graphs, leading to a new parameter, the capture time of a graph. While the capture time of a cop-win graph on n vertices is bounded above by n−3, half the number of vertices is sufficient for a large class of graphs including chordal graphs. Examples are given of cop-win graphs which have unique corners and have capture time within a small additive constant of the number of vertices. We consider the ratio of the capture time to the number of vertices, and extend this notion of capture time density to infinite graphs. For the infinite random graph, the capture time density can be any real number in [0,1]. We also consider the capture time when more than one cop is required to win. While the capture time can be calculated by a polynomial algorithm if the number k of cops is fixed, it is NP-complete to decide whether k cops can capture the robber in no more than t moves for every fixed t.     

The impact of loops on the game of cops and robbers on graphs

In the game of cops and robbers on graphs, the cops and the robber are allowed to pass their turn if they are located on a looped vertex. This paper explores the effect of loops on the cop number and the capture time. We provide examples of graphs where the cop number almost doubles when the loops are removed, graphs where the cop number decreases when the loops are removed, graphs where the capture time is quadratic in the number of vertices and copwin graphs where the cop needs to move away from the robber in optimal play.     

Meyniel's conjecture holds for random d‐regular graphs

In the game of cops and robber, the cops try to capture a robber moving on the vertices of the graph. The minimum number of cops required to win on a given graph G is called the cop number of G. The biggest open conjecture in this area is the one of Meyniel, which asserts that for some absolute constant C, the cop number of every connected graph G is at most . In a separate paper, we showed that Meyniel's conjecture holds asymptotically almost surely for the binomial random graph. The result was obtained by showing that the conjecture holds for a general class of graphs with some specific expansion‐type properties. In this paper, this deterministic result is used to show that the conjecture holds asymptotically almost surely for random d‐regular graphs when d = d(n) ≥ 3.     

On a pursuit game on cayley graphs

The game cops and robbers is considered on Cayley graphs of abelian groups. It is proved that if the graph has degreed, then [(d+1)/2] cops are sufficient to catch one robber. This bound is often best possible.     

The capture time of grids

We consider the game of cops and robber played on the Cartesian product of two trees. Assuming the players play perfectly, it is shown that if there are two cops in the game, then the length of the game (known as the 2-capture time of the graph) is equal to half the diameter of the graph. In particular, the 2-capture time of the m×n grid is proved to be .     

A Note on the Cops and Robber Game on Graphs Embedded in Non-Orientable Surfaces

We consider the two-player, complete information game of Cops and Robber played on undirected, finite, reflexive graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph. Let c(g) be the supremum over all cop numbers of graphs embeddable in a closed orientable surface of genus g, and likewise ${\tilde c(g)}$ for non-orientable surfaces. It is known (Andreae, 1986) that, for a fixed surface, the maximum over all cop numbers of graphs embeddable in this surface is finite. More precisely, Quilliot (1985) showed that c(g) ≤ 2g + 3, and Schröder (2001) sharpened this to ${c(g)\le \frac32g + 3}$ . In his paper, Andreae gave the bound ${\tilde c(g) \in O(g)}$ with a weak constant, and posed the question whether a stronger bound can be obtained. Nowakowski & Schröder (1997) obtained ${\tilde c(g) \le 2g+1}$ . In this short note, we show ${\tilde c(g) \leq c(g-1)}$ , for any g ≥ 1. As a corollary, using Schröder’s results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3, the maximum cop number of graphs embeddable in the Klein Bottle is at most 4, ${\tilde c(3) \le 5}$ , and ${\tilde c(g) \le \frac32g + 3/2}$ for all other g.     

Bounds on the length of a game of Cops and Robbers

In the game of Cops and Robbers, a team of cops attempts to capture a robber on a graph $G$. All players occupy vertices of $G$. The game operates in rounds; in each round the cops move to neighboring vertices, after which the robber does the same. The minimum number of cops needed to guarantee capture of a robber on $G$ is the cop number of $G$, denoted $c\left(G\right)$, and the minimum number of rounds needed for them to do so is the capture time. It has long been known that the capture time of an $n$-vertex graph with cop number $k$ is $O\left(n^{k+1}\right)$. More recently, Bonato et al. (2009) and Gaven?iak (2010) showed that for $k=1$, this upper bound is not asymptotically tight: for graphs with cop number 1, the cop can always win within $n?4$ rounds. In this paper, we show that the upper bound is tight when $k\ge 2$: for fixed $k\ge 2$, we construct arbitrarily large graphs $G$ having capture time at least ${\left(\frac{\left|V\left(G\right)\right|}{40k^4}\right)}^{k+1}$.In the process of proving our main result, we establish results that may be of independent interest. In particular, we show that the problem of deciding whether $k$ cops can capture a robber on a directed graph is polynomial-time equivalent to deciding whether $k$ cops can capture a robber on an undirected graph. As a corollary of this fact, we obtain a relatively short proof of a major conjecture of Goldstein and Reingold (1995), which was recently proved through other means (Kinnersley, 2015). We also show that $n$-vertex strongly-connected directed graphs with cop number 1 can have capture time $\Omega \left(n^2\right)$, thereby showing that the result of Bonato et al. (2009) does not extend to the directed setting.     

Locating a robber on a graph

Consider the following game of a cop locating a robber on a connected graph. At each turn, the cop chooses a vertex of the graph to probe and receives the distance from the probe to the robber. If she can uniquely locate the robber after this probe, then she wins. Otherwise the robber may either stay put or move to any vertex adjacent to his location other than the probe vertex. The cop’s goal is to minimize the number of probes required to locate the robber, while the robber’s goal is to avoid being located. This is a synthesis of the cop and robber game with the metric dimension problem. We analyse this game for several classes of graphs, including cycles and trees.     

On a game of policemen and robber

We consider a game where policemen try to catch a robber on a graph G (as defined by A. Quilliot) and we find the exact minimal number of policemen needed when G is a Cartesian product of trees.     

Hypertree width and related hypergraph invariants

We study the notion of hypertree width of hypergraphs. We prove that, up to a constant factor, hypertree width is the same as a number of other hypergraph invariants that resemble graph invariants such as bramble number, branch width, linkedness, and the minimum number of cops required to win Seymour and Thomas’s robber and cops game.     

A note on k-cop-win graphs

The game of Cops and Robbers is a very well known game played on graphs. In this paper we will show that minimum order of a graph that needs $k$ cops to guarantee the robber’s capture is increasing in $k$.     

A witness version of the Cops and Robber game

The games considered are mixtures of Searching and Cops and Robber. The cops have partial information provided via witnesses who report “sightings” of the robber. The witnesses are able to provide information about the robber’s position but not the direction in which he is moving. The robber has perfect information. In the case when sightings occur at regular intervals, we present a recognition theorem for graphs on which a single cop suffices to guarantee a win. In a special case, this recognition theorem provides a characterization.     

Directed Path-width and Monotonicity in Digraph Searching

Directed path-width was defined by Reed, Thomas and Seymour around 1995. The author and P. Hajnal defined a cops-and-robber game on digraphs in 2000. We prove that the two notions are closely related and for any digraph D, the corresponding graph parameters differ by at most one. The result is achieved using the mixed-search technique developed by Bienstock and Seymour. A search is called monotone, in which the robber's territory never increases. We show that there is a mixed-search of D with k cops if and only if there is a monotone mixed-search with k cops. For our cops-and-robber game we get a slightly weaker result: the monotonicity can be guaranteed by using at most one extra cop. On leave from Bolyai Institute, University of Szeged, Hungary. This research has been supported by a Marie Curie Fellowship of the European Community under contract number HPMF-CT-2002-01868 and by OTKA Grants F.030737 and T.34475.     

Edge critical cops and robber

We consider edge critical graphs when playing cops and robber. Specifically, we look at those graphs whose copnumbers change from one to two when any edge is added, deleted, subdivided or contracted. We characterize all such sets, showing that they are empty, trees, all 2-edge-connected graphs and empty, respectively. We also consider those graphs which change from copnumber two to one when any edge is added, and give a characterization in the k-regular case.     

Variations on cops and robbers

We consider several variants of the classical Cops and Robbers game. We treat the version where the robber can move R≥1 edges at a time, establishing a general upper bound of , where α = 1 + 1/R, thus generalizing the best known upper bound for the classical case R = 1 due to Lu and Peng, and Scott and Sudakov. We also show that in this case, the cop number of an n‐vertex graph can be as large as n^{1 ? 1/(R ? 2)} for finite R≥5, but linear in n if R is infinite. For R = 1, we study the directed graph version of the problem, and show that the cop number of any strongly connected digraph on n vertices is O(n(loglogn)²/logn). Our approach is based on expansion. © 2011 Wiley Periodicals, Inc. J Graph Theory.