Number Of Cops Research Articles

Cops-Robber Games and the Resolution of Tseitin Formulas

We characterize several complexity measures for the resolution of Tseitin formulas in terms of a two person cop-robber game. Our game is a slight variation of the one Seymour and Thomas used in order to characterize the tree-width parameter. For any undirected graph, by counting the number of cops needed in our game in order to catch a robber in it, we are able to exactly characterize the width, variable space, and depth measures for the resolution of the Tseitin formula corresponding to that graph. We also give an exact game characterization of resolution variable space for any formula. We show that our game can be played in a monotone way. This implies that the associated resolution measures on Tseitin formulas correspond exactly to those under the restriction of Davis-Putnam resolution, implying that this kind of resolution is optimal on Tseitin formulas for all the considered measures. Using our characterizations, we improve the existing complexity bounds for Tseitin formulas showing that resolution width, depth, and variable space coincide up to a logarithmic factor, and that variable space is bounded by the clause space times a logarithmic factor.

Variations of cops and robbers game on grids

Cops and robber is a two player turn-based game played on a graph where cops try to capture the robber. The cop number of a graph is the minimum number of cops required to capture the robber. We consider two variants of this game, namely cops and attacking robber and lazy cops and robber. In cops and attacking robber version, the robber can attack a cop and remove that cop from the game, whereas in lazy cops and robber version only one cop is allowed to move on a cop’s turn. We investigate the cop number of various grids (planar, cylindrical, toroidal and high dimension) and hypercubes for these variants.

Containment: A Variation of Cops and Robber

We consider “Containment”: a variation of the graph pursuit game of Cops and Robber in which cops move from edge to adjacent edge, the robber moves from vertex to adjacent vertex (but cannot move along an edge occupied by a cop), and the cops win by “containing” the robber—that is, by occupying all $$\deg (v)$$ of the edges incident with a vertex v while the robber is at v. We develop bounds that relate the minimal number of cops, $$\xi (G)$$, required to contain a robber to the well-known “cop-number” c(G) in the original game: in particular, $$c(G) {\le } \xi (G) {\le } \gamma (G) \Delta (G)$$. We note that $$\xi (G) {\ge } \Delta (G)$$ for all graphs G, and analyze several families of graphs in which equality holds, as well as several in which the inequality is strict. We also give examples of graphs which require an unbounded number of cops in order to contain a robber, and show that there exist cubic graphs on n vertices with $$\xi (G) = \Omega (n^{1/6})$$.

Limited visibility Cops and Robber

We consider a variation of the Cops and Robber game where the cops can only see the robber when the distance between them is at most a fixed parameter ℓ. We consider the basic consequences of this definition for some simple graph families, and show that this model is not monotonic, unlike common models where the robber is invisible. We see that cops’ strategy consists of a phase in which they need to “see” the robber (move within distance ℓ of the robber), followed by a phase in which they capture the robber. In some graphs the first phase is the most resource intensive phase (in terms of number of cops needed), while in other graphs, it is the second phase. Finally, we characterize those trees for which k cops are sufficient to guarantee capture of the robber for all ℓ≥1.

Comparing the power of cops to zombies in pursuit-evasion games

We compare two kinds of pursuit-evasion games played on graphs. In Cops and Robbers, the cops can move strategically to adjacent vertices as they please, while in a new variant, called deterministic Zombies and Survivors, the zombies (the counterpart of the cops) are required to always move towards the survivor (the counterpart of the robber). The cop number of a graph is the minimum number of cops required to catch the robber on that graph; the zombie number of a graph is the minimum number of zombies required to catch the survivor on that graph. We answer two questions from the 2016 paper of Fitzpatrick, Howell, Messinger, and Pike. We show that for any integers m and k such that m≥k≥1, there is a graph with zombie number m and cop number k. We also show that the zombie number of the n-dimensional hypercube is ⌈2n∕3⌉.

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.

A proof of the Meyniel conjecture for Abelian Cayley graphs

We prove that the cop number of a connected abelian Cayley graph on n vertices is bounded by 7n. This proves that H. Meyniel’s conjectured bound of O(n) for the cop number of any connected graph on n vertices holds for abelian Cayley graphs.

Selfish cops and active robber: Multi-player pursuit evasion on graphs

We introduce and study the game of “Selfish Cops and Active Robber” (SCAR) which can be seen as a multiplayer variant of the “classic” two-player Cops and Robbers (CR) game. In classic CR all cops are controlled by a single player, who has no preference over which cop captures the robber. In SCAR, on the other hand, each of N−1 cops is controlled by a separate player, and a single robber is controlled by the N-th player; and the capturing cop player receives a higher reward than the non-capturing ones. Consequently, SCAR is an N-player pursuit game on graphs, in which each cop player has an increased motive to be the one who captures the robber. The focus of our study is the existence and properties of SCAR Nash Equilibria (NE). In particular, we prove that SCAR always has one NE in deterministic positional strategies and (for N≥3) another one in, generally, deterministic nonpositional strategies. Furthermore, we study conditions which, at equilibrium, guarantee either capture or escape of the robber and show that (because of the antagonism between the “selfish” cop players) the robber may, in certain SCAR configurations, be captured later than he would be in classic CR, or even not captured at all. Finally we define the selfish cop number of a graph and study its connection to the classic cop number.

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.

Generalized Cops and Robbers: A Multi-player Pursuit Game on Graphs

We introduce and study the Generalized Cops and Robbers (GCR) game, an N-player pursuit game in graphs. The two-player version is essentially equivalent to the classic Cops and Robbers (CR) game. The three-player version can be understood as two CR games played simultaneously on the same graph; a player can be at the same time both pursuer and evader. The same is true for four or more players. We formulate GCR as a discounted stochastic game of perfect information and prove that, for three or more players, it has at least two Nash equilibria: one in positional deterministic strategies and another in nonpositional ones. We also study the capturing properties of GCR Nash equilibria in connection with the cop number of a graph. Finally, we briefly discuss GCR as a member of a wider family of multi-player graph pursuit games with rather interesting properties.

Hyperopic Cops and Robbers

We introduce a new variant of the game of Cops and Robbers played on graphs, where the robber is invisible unless outside the neighbor set of a cop. The hyperopic cop number is the corresponding analogue of the cop number, and we investigate bounds and other properties of this parameter. We characterize the cop-win graphs for this variant, along with graphs with the largest possible hyperopic cop number. We analyze the cases of graphs with diameter 2 or at least 3, focusing on when the hyperopic cop number is at most one greater than the cop number. We show that for planar graphs, as with the usual cop number, the hyperopic cop number is at most 3. The hyperopic cop number is considered for countable graphs, and it is shown that for connected chains of graphs, the hyperopic cop density can be any real number in [0,1/2].

Localization game on geometric and planar graphs

The main topic of this paper is motivated by a localization problem in cellular networks. Given a graph G we want to localize a walking agent by checking his distance to as few vertices as possible. The model we introduce is based on a pursuit graph game that resembles the famous Cops and Robbers game. It can be considered as a game theoretic variant of the metric dimension of a graph. We provide upper bounds on the related graph invariant ζ(G), defined as the least number of cops needed to localize the robber on a graph G, for several classes of graphs (trees, bipartite graphs, etc.). Our main result is that, surprisingly, there exists planar graphs of treewidth 2 and unbounded ζ(G). On a positive side, we prove that ζ(G) is bounded by the pathwidth of G. We then show that the algorithmic problem of determining ζ(G) is NP-hard in graphs with diameter at most 2. Finally, we show that at most one cop can approximate (arbitrary close) the location of the robber in the Euclidean plane.

A Survey on the Relationship Between the Game of Cops and Robbers and Other Game Representations

This paper surveys the similarities and differences between the game of cops and robbers and classes of related games. The game of cops and robbers is a class of pursuit–evasion games played on a finite graph. The players comprise a set of cops and a robber located on vertices of the graph, and they move alternately along edges of the graph. The cops win the game if at least one cop occupies the same vertex of the robber in finite time; otherwise, the robber wins. Quilliot (Jeux de points fixes sur les graphes, These de 3ieme Cycle, Universite de Paris VI, Paris, pp 131–145, 1978) and Nowakowski and Winkler (Discrete Math 43:235–239, 1983) were among the first to study this class of games. The main questions studied in the literature of cops and robbers games are on the class of graphs that characterize cop-win games, on the minimum number of cops needed for classes of graphs to be cop-win, on a conjecture due Meyniel (Personal communication, 1985) about the cop number of a graph and more recently on the game played on random and/or infinite graphs (Bonato and Nowakowski in The game of cops and robbers, Student mathematical library, vol 61, American Mathematical Society, Providence, 2010; Bonato and Pralat in Graph searching games and probabilistic methods, CRC Press, Boca Raton, 2017). While the game of cops and robbers is not traditionally modeled in the language of extensive game theory (Kuhn, in: Kuhn, Tucker (eds) Contributions to the theory of games II, Princeton University Press, Princeton, 1953), Kehagias et al. (The role of visibility in pursuit/evasion games, 2014) give a general framework for the game of cops and robbers that relate to extensive games. By using the notion of “capture time,” they are able to represent the game using a standard discrete dynamic game model with explicit payoff functions for the cops and the robber. More recently, the game of cops and robbers has also been related to dynamic games with state variables (Bonato and Macgillivray in Contrib Discrete Math, 2017) and discounted stochastic games (Konstantinidis and Kehagias in Selfish cops and adversarial robber: multi-player pursuit evasion on graphs, 2017a, arXiv:1703.07695 ; Theor Comput Sci 680:25–35, 2017b). In this survey, we study these connections.

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(G), 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(nk+1). More recently, Bonato et al. (2009) and Gavenčiak (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≥2: for fixed k≥2, we construct arbitrarily large graphs G having capture time at least V(G)40k4k+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 Ω(n2), thereby showing that the result of Bonato et al. (2009) does not extend to the directed setting.

Throttling for the game of Cops and Robbers on graphs

We consider the cop-throttling number of a graph G for the game of Cops and Robbers, which is defined to be the minimum of (k+captk(G)), where k is the number of cops and captk(G) is the minimum number of rounds needed for k cops to capture the robber on G over all possible games. We provide some tools for bounding the cop-throttling number, including showing that the positive semidefinite (PSD) throttling number, a variant of zero forcing throttling, is an upper bound for the cop-throttling number. We also characterize graphs having low cop-throttling number and investigate how large the cop-throttling number can be for a given graph. We consider trees, unicyclic graphs, incidence graphs of finite projective planes (a Meyniel extremal family of graphs), a family of cop-win graphs with maximum capture time, grids, and hypercubes. All the upper bounds on the cop-throttling number we obtain for families of graphs are O(n).

Cops and Robbers on intersection graphs

The cop number of a graph G is the smallest k such that k cops win the game of cops and robber on G. We investigate the maximum cop number of geometric intersection graphs, which are graphs whose vertices are represented by geometric shapes and edges by their intersections. We establish the following dichotomy for previously studied classes of intersection graphs: •The intersection graphs of arc-connected sets in the plane (called string graphs) have cop number at most 15, and more generally, the intersection graphs of arc-connected subsets of a surface have cop number at most 10g+15 in case of orientable surface of genus g, and at most 10g′+15 in case of non-orientable surface of Euler genus g′. For more restricted classes of intersection graphs, we obtain better bounds: the maximum cop number of interval filament graphs is two, and the maximum cop number of outer-string graphs is between 3 and 4.•The intersection graphs of disconnected 2-dimensional sets or of 3-dimensional sets have unbounded cop number even in very restricted settings. For instance, it follows from known results that the cop number is unbounded on intersection graphs of two-element subsets of a line. We further show that it is also unbounded on intersection graphs of 3-dimensional unit balls, of 3-dimensional unit cubes or of 3-dimensional axis-aligned unit segments.

On Cops and Robbers on $G^{\Xi}$ and cop-edge critical graphs

Cop Robber game is a two player game played on an undirected graph. In this game cops try to capture a robber moving on the vertices of the graph. The cop number of a graph is the least number of cops needed to guarantee that the robber will be caught. In this paper we presents results concerning games on $G^{\Xi}$, that is the graph obtained by connecting the corresponding vertices in $G$ and its complement $\overline{G}$. In particular we show that for planar graphs $c(G^{\Xi})\leq 3$. Furthermore we investigate the cop-edge critical graphs, i.e. graphs that for any edge $e$ in $G$ we have either $c(G-e) c(G)$. We show couple examples of cop-edge critical graphs having cop number equal to~$3$.

Catching a fast robber on the grid

We study the problem of cops and robbers on the grid where the robber is allowed to move faster than the cops. It is well known that two cops are necessary and sufficient to catch the robber on any finite grid when the robber has unit speed. Here, we prove that if the speed of the robber exceeds a sufficiently large absolute constant, then the number of cops needed to catch the robber on an n×n grid is exp⁡(Ω(log⁡n/log⁡log⁡n)).

Linguistic geometry approach for solving the Cops and Robber problem in grid environments

The Cops and Robber problem is a game played between two adversary groups, in which one group (cops) attempts to capture the other group (robber). In this paper, the optimal capture time problem is addressed, which seeks the best motion strategy from the viewpoint of the cops for capturing the robber within the minimum time or number of steps, and a new algorithm based on linguistic geometry (LG) is proposed. LG is widely used for solving two-player abstract board games and wargaming problems by breaking the system (problem) into several subsystems, in each of which the playing elements and their paths (called ‘trajectories’) are identified, and a network consisting of these trajectories is constructed. A solution is identified by searching the subsystems by the alpha-beta pruning method. Also, to provide a lower bound for the robber's capture time, we have enhanced the alpha-beta pruning method by incorporating a new admissible heuristic more informed than existing heuristics. As a result, the branching factor of the search is decreased, and the problem is solved more efficiently. For experimentation, several Cops and Robber problems on regular and holed square grids were solved by the LG-based method, and the results were compared with the Reverse Minimax A* (RMA*) algorithm. According to the results, the presented algorithm requires far less computational time and memory than the RMA* method (in fact, approximately 95.5% and 97.5% are saved on average, respectively), and near-optimal solutions were obtained. Also, it was observed that while increasing the number of cops severely affects the computational time and required memory of the RMA*, it has insignificant effects on our algorithm.

The Game of Overprescribed Cops and Robbers Played on Graphs

We consider the effect on the length of the game of Cops and Robbers when more cops are added to the game play. In Overprescribed Cops and Robbers, as more cops are added, the capture time (the minimum length of the game assuming optimal play) monotonically decreases. We give the full range of capture times for any number of cops on trees, and classify the capture time for an asymptotic number of cops on grids, hypercubes, and binomial random graphs. The capture time of planar graphs with a number of cops at and far above the cop number is considered.