Number Of Cops Research Articles

Compact Metric Spaces with Infinite Cop Number

AbstractMohar recently adapted the classical game of Cops and Robber from graphs to metric spaces, thereby unifying previously studied pursuit-evasion games. He conjectured that finitely many cops can win on any compact geodesic metric space, and that their number can be upper-bounded in terms of the ranks of the homology groups when the space is a simplicial pseudo-manifold. We disprove these conjectures by constructing a metric on $$\mathbb {S}^3$$ S 3 with infinite cop number. More problems are raised than settled.

Improved bounds on the cop number when forbidding a minor

AbstractAndreae proved that the cop number of connected ‐minor‐free graphs is bounded for every graph . In particular, the cop number is at most if contains no isolated vertex, where . The main result of this paper is an improvement on this bound, which is most significant when is small or sparse, for instance, when can be obtained from another graph by multiple edge subdivisions. Some consequences of this result are improvements on the upper bound for the cop number of ‐minor‐free graphs, ‐minor‐free graphs and linklessly embeddable graphs.

A cops and robber game and the meeting time of synchronous directed walks

AbstractIn a previous work, the authors showed that the maximum number of infinitely long synchronous directed walks that never meet is equal to the dimension of the no‐meet matroid, namely the largest order of a collection of vertex‐disjoint cycles. Given , we want to compute the meeting time of walks: the first time step such that, given any set of walks, at least two of them must meet no later than . We precisely prove that the meeting time is at most , where is the number of vertices. A connection is established with a cops and robber game on directed graphs with helicopter cops and an invisible slow robber. The meeting time of walks equals the capture time in this game, when at most capture attempts are allowed. While this capture time can be computed in polynomial time, we show that it is NP‐hard to compute the minimum number of cops needed to catch the robber. More insights are also given on the number and its relation to pathwidth and other graph parameters. Finally we analyze these game measures on digraph tensor products.

Catching a robber on a random k-uniform hypergraph

Abstract The game of Cops and Robber is usually played on a graph, where a group of cops attempt to catch a robber moving along the edges of the graph. The cop number of a graph is the minimum number of cops required to win the game. An important conjecture in this area, due to Meyniel, states that the cop number of an n-vertex connected graph is $O(\sqrt {n})$ . In 2016, Prałat and Wormald showed that this conjecture holds with high probability for random graphs above the connectedness threshold. Moreover, Łuczak and Prałat showed that on a $\log $ -scale the cop number demonstrates a surprising zigzag behavior in dense regimes of the binomial random graph $G(n,p)$ . In this paper, we consider the game of Cops and Robber on a hypergraph, where the players move along hyperedges instead of edges. We show that with high probability the cop number of the k-uniform binomial random hypergraph $G^k(n,p)$ is $O\left (\sqrt {\frac {n}{k}}\, \log n \right )$ for a broad range of parameters p and k and that on a $\log $ -scale our upper bound on the cop number arises as the minimum of two complementary zigzag curves, as opposed to the case of $G(n,p)$ . Furthermore, we conjecture that the cop number of a connected k-uniform hypergraph on n vertices is $O\left (\sqrt {\frac {n}{k}}\,\right )$ .

Cops and Robber on butterflies, grids, and AT-free graphs

Cops and Robber is a well-studied two player pursuit-evasion game played on a graph. In this game, a set of cops, controlled by the first player, tries to capture the position of a robber, controlled by the second player. The cop number of a graph is the minimum number of cops required to capture the robber in the graph. A group of cops guard a subgraph if they can ensure that the robber cannot enter the subgraph without getting captured immediately.We study the applications of guarding to provide new bounds and improve existing bounds for several graph classes. In particular, we show that the cop number for butterfly networks and for solid grids is two. We also construct a partial grid with cop number 3 establishing that partial grids have same cop number as their superclass planar graphs. We also consider three well-studied variants of Cops and Robber: Cops and Fast Robber, Cops and Attacking Robber, and Surrounding CnR. We improve the existing bounds for the cop number of multidimensional grids for both Cops and Attacking Robber and Surrounding CnR. Finally, we consider Cops and Fast Robber on AT-free graphs to improve the existing bounds on the cop number for this game.

Coarse geometry of the cops and robber game

We introduce two variations of the cops and robber game on graphs. These games yield two invariants in Z+∪{∞} for any connected graph Γ, the weak cop numberwCop(Γ) and strong cop numbersCop(Γ). These invariants satisfy that sCop(Γ)≤wCop(Γ). Any graph that is finite or a tree has strong cop number one. These new invariants are preserved under small local perturbations of the graph, specifically, both the weak and strong cop numbers are quasi-isometric invariants of connected graphs. More generally, we prove that if Δ is a quasi-retract of Γ then wCop(Δ)≤wCop(Γ) and sCop(Δ)≤sCop(Γ). We exhibit families of examples of graphs with arbitrary weak cop numbers (resp. strong cop number). We prove that hyperbolic graphs have strong cop number one. We also prove that one-ended non-amenable locally-finite vertex-transitive graphs have infinite weak cop number. We raise the question of whether there exists a connected vertex transitive graph with finite weak (resp. strong) cop number different than one.

On the cop number of Sierpinski-like graphs

In this study, the cops and robber game is transferred to the Sierpinski graph, Sierpinski-like graphs [Formula: see text] and [Formula: see text], Sierpinski gasket graph [Formula: see text], and generalized Sierpinski graphs [Formula: see text] where [Formula: see text] has an order four and [Formula: see text]. We show that the cop number of these graphs is [Formula: see text], excluding [Formula: see text]. We also give a strategy for the cops to win.

Cops and robber on subclasses of P5-free graphs

The game of cops and robber is a turn-based vertex pursuit game played on a connected graph between a team of cops and a single robber. The cops and the robber move alternately along the edges of the graph. We say that the team of cops wins the game if a cop and the robber are at the same vertex of the graph. The minimum number of cops required to win in a connected graph is called the cop number of the graph. Sivaraman (2019) [16] conjectured that for every t≥5, the cop number of a connected Pt-free graph is at most t−3, where Pt denotes a path on t vertices. Turcotte (2022) [18] showed that the cop number of any 2K2-free graph is at most 2, which was earlier conjectured by Sivaraman and Testa. Note that if a connected graph is 2K2-free, then it is also P5-free. Liu showed that the cop number of a connected (Pt, H)-free graph is at most t−3, where H is a cycle of length at most t or a claw. So the conjecture of Sivaraman is true for (P5, H)-free graphs, where H is a cycle of length at most 5 or a claw. In this paper, we show that the cop number of a connected (P5,H)-free graph is at most 2, where H∈{diamond, paw, K4, 2K1∪K2, K3∪K1}.

Pursuit-evasion games on Latin square graphs

We investigate various pursuit-evasion parameters on latin square graphs, including the cop number, metric dimension, and localization number. The cop number of latin square graphs is studied, and for $k$-MOLS$(n),$ bounds for the cop number are given. If $n>(k+1)^2,$ then the cop number is shown to be $k+2.$ Lower and upper bounds are provided for the metric dimension and localization number of latin square graphs. The metric dimension of back-circulant latin squares shows that the lower bound is close to tight. Recent results on covers and partial transversals of latin squares provide the upper bound of $n+O\left(\frac{\log{n}}{\log{\log{n}}}\right)$ on the localization number of a latin square graph of order $n.$

Cops and Robbers: A Game of Pursuit on Graphs

Cops and Robbers is an exciting game that can be mathematically modelled and investigated with the use of graph theory, counting principles and probability. A discrete version of this game is investigated in the present study. A Cop and a Robber are two opponents placed upon the battlefield. The goal of the Cop is to capture the Robber by landing on the same vertex as him. The goal of the Robber is to avoid the Cop. Simple graph examples are used to showcase the principles of this discrete game. The notion of pitfalls for the Robber is introduced, and their role in determining whether a graph is a Cop-win or Robber-win is explained. Then, more difficult graph examples are presented together with variations in the number of Cops on the battlefield and the number of ways the Cops and the Robbers can travel upon a graph are calculated. Finally, the probability of a Cop and Robber meeting on a specific vertex is found for a three-dimensional graph.

Cops and robber on variants of retracts and subdivisions of oriented graphs (Brief Announcement)

Cops and Robber is one of the most studied two-player pursuit-evasion games played on graphs, where multiple cops, controlled by one player, pursue a single robber. The cop number of a graph is the minimum number of cops that can ensure the capture of the robber. In directed graphs, two kinds of moves are defined for players: strong move, where a player can move both along and against the orientation of an arc to an adjacent vertex; and weak move, where a player can only move along the orientation of an arc to an out-neighbor. We study three variants of Cops and Robber on oriented graphs: strong cop model, where the cops can make strong moves while the robber can only make weak moves; normal cop model, where both cops and the robber can only make weak moves; and weak cop model, where the cops can make weak moves while the robber can make strong moves. We study the cop number of these models with respect to several variants of retracts on oriented graphs and establish that the strong and normal cop number of an oriented graph remains invariant in their strong and distributed retracts, respectively. Next, we go on to study all three variants with respect to the subdivisions of graphs and oriented graphs. Finally, we establish that all these variants remain computationally difficult even when restricted to the class of 2-degenerate bipartite graphs.

A Note on Some Weaker Notions of Cop-Win and Robber-Win Graphs

The game of pursuit and evasion, when played on graphs, is often referred to as the game of cops and robbers. This classical version of the game has been completely solved by Nowakowski and Winkler, who gave the exact class of graphs for which the pursuer can win the game (cop-win). When the graph does not satisfy the dismantlability property, Nowakowski and Winkler’s Theorem does not apply. In this paper, we give some weaker notions of cop-win and robber-win graphs. In particular, we fix the number of cops to be equal to one, and we ask whether there exist sets of initial conditions for which the graph can be cop-win or robber-win. We propose some open questions related to this initial condition problem with the goal of developing both structural characterizations and algorithms that are computable in polynomial time (P) and that can solve weakly cop-win and weakly- robber-win graphs.

A note on cops and robbers, independence number, domination number and diameter

We study relationships between the diameter D(G), the domination number γ(G), the independence number α(G) and the cop number c(G) of a connected graph G, showing (i.) c(G)≤α(G)−⌊D(G)−32⌋, and (ii.) c(G)≤γ(G)−D(G)3+O(D(G)).

A note on hyperopic cops and robber

In this paper, we explore a variant of the game of Cops and Robber introduced by Bonato et al. where the robber is invisible unless outside the common neighborhood of the cops. The hyperopic cop number is analogous to the cop number and we investigate bounds on this quantity. We define a small common neighborhood set and relate the minimum cardinality of this graph parameter to the hyperopic cop number. We consider diameter [Formula: see text] graphs, particularly the join of two graphs, as well as Cartesian products.

The Game of Cops and Robbers on Directed Graphs with Forbidden Subgraphs

The traditional game of cops and robbers is played on undirected graph. Recently, the same game played on directed graph is getting attention by more and more people. We knew that if we forbid some subgraph we can bound the cop number of the corresponding class of graphs. In this paper, we analyze the game of cops and robbers on $\Vec{H}$-free digraphs. However, it is not the same as the case of undirected graph. So we give a new concept ($\Vec{H}^*$-free) to get a similar conclusion about the case of undirected graph.

On the cop number of graphs of high girth

AbstractWe establish a lower bound for the cop number of graphs of high girth in terms of the minimum degree, and more generally, in terms of a certain growth condition. We show, in particular, that the cop number of any graph with girth and minimum degree is at least . We establish similar results for directed graphs. While exposing several reasons for conjecturing that the exponent in this lower bound cannot be improved to , we are also able to prove that it cannot be increased beyond . This is established by considering a certain family of Ramanujan graphs. In our proof of this bound, we also show that the “weak” Meyniel's conjecture holds for expander graph families of bounded degree.

One-visibility cops and robber on trees: Optimal cop-win strategies

In the one-visibility cops and robber game on a graph, the robber is visible to the cops only when the robber is in the closed neighbourhood of the vertices occupied by the cops. The one-visibility copnumber of a graph is the minimum number of cops required to capture the robber on the graph. In this paper, we investigate the one-visibility cops and robber game on trees. For trees, we introduce a key structure, called road, for characterising optimal cop-win strategies. We give an O(nlog⁡n) time algorithm to compute an optimal cop-win strategy for a tree with n vertices. We also establish relations between zero-visibility and one-visibility copnumbers on trees.

Lazy Cops and Robber on Certain Cayley Graph

The lazy cop number is the minimum number of cops needed for the cops to have a winning strategy in the game of Cops and Robber where at most one cop may move in any one round. This variant of the game of Cops and Robber, called Lazy Cops and Robber, was introduced by Offner and Ojakian, who provided bounds for the lazy cop number of hypercubes. In this paper, we are interested in the game of Lazy Cops and Robber on a Cayley graph of n copies of Zm+1.

On generalised Petersen graphs of girth 7 that have cop number 4

We show that if n = 7k/i with i ∈ {1, 2, 3} then the cop number of the generalised Petersen graph GP(n,k) is 4, with some small previously-known exceptions. It was previously proved by Ball et al. (2015) that the cop number of any generalised Petersen graph is at most 4. The results in this paper explain all of the known generalised Petersen graphs that actually have cop number 4 but were not previously explained by Morris et al. in a recent preprint, and places them in the context of infinite families. (More precisely, the preprint by Morris et al. explains all known generalised Petersen graphs with cop number 4 and girth 8, while this paper explains those that have girth 7.)

Meyniel Extremal Families of Abelian Cayley Graphs

We study the game of Cops and Robbers, where cops try to capture a robber on the vertices of a graph. Meyniel’s conjecture states that for every connected graph G on n vertices, the cop number of G is upper bounded by $$O(\sqrt{n})$$ . That is, for every graph G on n vertices $$O(\sqrt{n})$$ cops suffice to catch the robber. We present several families of abelian Cayley graphs that are Meyniel extremal, i.e., graphs whose cop number is $$O(\sqrt{n})$$ . This proves that the $$O(\sqrt{n})$$ upper bound for Cayley graphs proved by Bradshaw (Discret Math 343:1, 2019) is tight. In particular, this shows that Meyniel’s conjecture, if true, is tight even for abelian Cayley graphs. In order to prove the result, we construct Cayley graphs on n vertices with $$\Omega (\sqrt{n})$$ generators that are $$K_{2,3}$$ -free. This shows that the Kövári, Sós, and Turán theorem, stating that any $$K_{2,3}$$ -free graph of n vertices has at most $$O(n^{3/2})$$ edges, is tight up to a multiplicative constant even for abelian Cayley graphs.