Robber Game Research Articles

Cops and robbers game and applications of its variant

Pursuit-evasion game is a game with a pursuer player and an evader player, it has numerous applications including artificial intelligence, robot motion planning and so on. The cops and robbers game is a type of pursuit-evasion game played on graphs, and its variant is a very fruitful research area in graph theory. The purpose of this paper is to find the specific relations between pursuit-evasion game applications and different types of variants of the cops and robbers game. The research method is to find variants with different game rules and winning strategies and classify them. Then compare these rules and strategies with the applications of pursuit-evasion games to find their relationships. The comparison result shows that there are many differences between the variants and the applications, which lead to their inability to be directly related. The application of the pursuit-evasion game is mainly based on route planning and object distance, and generally contains multiple variant types, which is different from the current research direction of variant.

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.

The zero-visibility cops and robber game on graph products

The zero-visibility cops and robber game is a variant of Cops and Robbers subject to the constraint that the cops have no information at any time about the location of the robber. We first study a partition problem in which for a given graph and an integer k, we want to find a partition of the vertex set such that the size of the boundary of the smaller subset in the partition is at most k while the size of this subset is as large as possible under some conditions. Then we apply such partitions to prove lower bounds on the zero-visibility cop numbers of graph products. We also investigate the monotonic zero-visibility cop numbers of graph products.

A cop and robber game on edge-periodic temporal graphs

We introduce a cops and robbers game with one cop and one robber on a special type of time-varying graphs (TVGs), namely edge-periodic graphs. These are TVGs in which, for each edge e, a binary string τ(e) is given such that the edge e is present in time step t if and only if τ(e) contains a 1 at position tmod|τ(e)|. This periodicity allows for a compact representation of infinite TVGs. We prove that even for very simple underlying graphs, i.e., directed and undirected cycles, the problem of deciding whether a cop-winning strategy exists is NP-hard and W[1]-hard parameterized by the number of vertices. Furthermore, we show that this decision problem can be solved on general edge-periodic graphs in PSPACE. Finally, we present tight bounds on the minimum length of a directed or undirected cycle that guarantees the cycle to be robber-winning.

Tipsy cop and tipsy robber: Collisions of biased random walks on graphs

Introduced by Harris, Insko, Prieto Langarica, Stoisavljevic, and Sullivan, the tipsy cop and drunken robber is a variant of the cop and robber game on graphs in which the robber simply moves randomly along the graph, while the cop moves directed towards the robber some fixed proportion of the time and randomly the remainder. In this article, we adopt a slightly different interpretation of tipsiness of the cop and robber where we assume that in any round of the game there are four possible outcomes: a sober cop move, a sober robber move, a tipsy (uniformly random) move by the cop, and a tipsy (uniformly random) move by the robber. We study this tipsy cop and tipsy robber game on the infinite grid graph and on certain families of infinite trees including δ-regular trees and δ-regular trees rooted to a Δ-regular tree, where Δ≥δ. Our main results analyze strategies for the cop and robber on these graphs. We conclude with some directions for further study.

Police and robber game on infinite chessboard

It is considered two variants of the game "Policeman and a robber", on an infinite chessboard that is a graph giving a regular partition of the plane into squares. Heuristic and precise definitions of the concepts "the initial state is winning for the pursuer", and "the initial state is winning for the evader" are formulated. Then, criteria for determining if a given initial state is winning for the pursuer or for the evader is given.

Distributed mobile CEP for collaborative social computing

Smartphones have become highly personalized interfaces between people and the technology ecosystem around them. In this sense, they play a key role for a technology shift from the current Internet of Things to a future human-centric paradigm of an Internet of People, automatically adapting smart things and services to the preferences and context of their users. In this paper, we propose the use of Complex Event Processing (CEP) engines deployed in the users’ smartphones granting them context-awareness capabilities in order to react to external stimulus, and enabling them to interact both with smart things and services in the surroundings of the users. With that purpose, we have designed a communication architecture that interconnects CEP engines running on smartphones, providing a framework for building applications for Mobile-based Collaborative Social Computing (MCSC). For that, we make use of previous works of the authors with Digital Avatars, a framework which promotes the use of smartphones for inferring and sharing a unique digital avatar or virtual profile of each user. The resulting framework, which we have called Collaborative CEP, allows to implement complex interactions among users, and between them and the IoT, a common need in Collaborative Social Computing applications. We provide a proof of concept based on the implementation of a Cops and Robbers game to test the expressiveness and correct functioning of the framework, and we evaluate its performance and efficiency.

Cops and Robber Game on Infinite Chessboard

Cops and Robber Game on Infinite Chessboard

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.

A cop-winning strategy on strongly cop-win graphs

Nowakowski and Winkler [5] introduced a game on graphs where a cop tries to catch a robber; in particular they showed that the cop has a winning strategy on a (finite or infinite) graph if and only if the graph is dismantlable. In this paper, we prove a similar result for a game on a finite graph where the robber can travel through arbitrary many edges in a single move. The original dismantling procedure consists in deleting specific vertices one by one until reaching a single vertex. This procedure fails with a fast robber because by deleting vertices we remove possible escape ways for the robber, and hence possibly change the nature of the graph from robber-win to cop-win. To overcome this problem, we define a new dedicated light-dismantling procedure, which is a generalization of the classical one, with a broader application possibility, not only to the fast robber game. Instead of being deleted, vertices are darkened: dark vertices can be only visited by the robber, in passing. Hence we introduce undergrounded graphs, on which the game will be played. We define a monotonic winning strategy for the cop that ensures that she is progressing towards the robber, thus securing a growing area.

Capture times in the bridge-burning cops and robbers game

In this paper, we consider a variant of the cops and robbers game on a graph, introduced by Kinnersley and Peterson, in which every time the robber uses an edge, it is removed from the graph, known as bridge-burning cops and robbers. In particular, we study the maximum time it takes the cops to capture the robber.

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.

Spy game: FPT-algorithm, hardness and graph products

In the (s,d)-spy game over a graph G, k guards and one spy occupy some vertices of G and, at each turn, the spy may move with speed s (along at most s edges) and each guard may move along one edge. The spy and the guards may occupy the same vertices. The spy wins if she reaches a vertex at distance more than the surveilling distance d from every guard. This game was introduced by Cohen et al. (2016) [15] and is related to two well-studied games: Cops and Robber game and Eternal Dominating game. The guard number gns,d(G) is the minimum number of guards such that the guards have a winning strategy (of controlling the spy) in the graph G. In 2018, it was proved that deciding if the spy has a winning strategy is NP-hard for every speed s≥2 and distance d≥0. In this paper, we initiate the investigation of the guard number in grids and in graph products. We obtain a strict upper bound on the strong product of two general graphs and obtain examples with King grids that match this bound and other examples for which the guard number is smaller. We also obtain the exact value of the guard number in the lexicographical product of two general graphs for any distance d≥2. From the algorithmic point of view, we prove a positive result: if the number k of guards is fixed, the spy game is solvable in polynomial XP time O(n3k+2) for every speed s≥2 and distance d≥0. In other words, the spy game is XP when parameterized by the number of guards. This XP algorithm is used to obtain an FPT algorithm on the P4-fewness of the graph. As a negative result, we prove that the spy game is W[2]-hard even in bipartite graphs when parameterized by the number of guards, for every speed s≥2 and distance d≥0, extending the hardness result of Cohen et al. in 2018.

Noise controlling entangling operators in the quantum network implementation of cop and robber game

Noise controlling entangling operators in the quantum network implementation of cop and robber game

Chasing a Drunk Robber in Many Classes of Graphs

A cops and robbers game (CR) on graphs was originated in 1983 by Quilliot and by Nowakowski and Winkler independently. This game has been applied in many problems in the area of theoretical computer science such as information seeking, robot motion planning or network security as evidenced by many conferences and publications. The CR game has also been extensively studied and varied to many versions. In this paper, we focus on CR game under the condition that the robber performs a random walk. Namely, he drunkenly, or randomly, chooses his next move to escape the cop, while the cop plays optimally in order to minimize the expected drunk capture times{textit{dct}}(G) of a graph G. We apply the concepts of expected hitting time in Markov Chain and combinatorial technique to find {textit{dct}}(G) for graphs G that have been used in many applications which are cycles, complete multipartite graphs, grid graphs and prism graphs.

A Cop and Drunken Robber Game on n-Dimensional Infinite-Grid Graphs

A Cop and Drunken Robber (CDR) game is one variation of a famous combinatorial game, called Cops and Robbers, which has been extensively studied and applied in the area of theoretical and computer science as demonstrated by several conferences and publications. In this paper, for a natural number n, we present two strategies for a single cop to chase a drunken robber on n-dimensional infinite-grid graphs. Both strategies show that if the initial distance between the cop and the drunken robber is s, then the expected capture time is s+o(s).

One-visibility cops and robber on trees

The one-visibility cops and robber game is a variation of the classic cops and robber game, where one-visibility means that the information of the robber is known to all cops only when the distance between the robber and at least one cop is at most one. In this paper, we give a lower bound on the one-visibility copnumber of general trees. We present strategies to clear trees according to their structures. We propose a linear-time algorithm for computing the one-visibility copnumber of trees.

General Cops and Robbers games with randomness

Cops and Robbers games have been studied for the last few decades in computer science and mathematics. As in general pursuit evasion games, pursuers (cops) seek to capture evaders (robbers); however, players move in turn and are constrained to move on a discrete structure, usually a graph, and know the exact location of their opponent. In 2017, Bonato and MacGillivray [2] presented a general characterization of Cops and Robbers games in order for them to be globally studied. However, their model doesn't cover cases where stochastic events may occur, such as the robbers moving in a random fashion. In this paper we present a novel model with stochastic elements that we call a Generalized Probabilistic Cops and Robbers game (GPCR). A typical such game is one where the robber moves according to a probabilistic distribution, either because she is rather lost or drunk than evading, or because she is a robot. We present results to solve GPCR games, thus enabling one to study properties relating to the optimal strategies in large classes of Cops and Robbers games. Some classic Cops and Robbers games properties are also extended.

TIPSY COP AND DRUNKEN ROBBER: A VARIANT OF THE COP AND ROBBER GAME ON GRAPHS

Motivated by a biological scenario illustrated in the YouTube video \url{ this https URL} where a neutrophil chases a bacteria cell moving in random directions, we present a variant of the cop and robber game on graphs called the tipsy cop and drunken robber game. In this game, we place a tipsy cop and a drunken robber at different vertices of a finite connected graph $G$. The game consists of independent moves where the robber begins the game by moving to an adjacent vertex from where he began, this is then followed by the cop moving to an adjacent vertex from where she began. Since the robber is inebriated, he takes random walks on the graph, while the cop being tipsy means that her movements are sometimes random and sometimes intentional. Our main results give formulas for the probability that the robber is still free from capture after $m$ moves of this game on highly symmetric graphs, such as the complete graphs, complete bipartite graphs, and cycle graphs. We also give the expected encounter time between the cop and robber for these families of graphs. We end the manuscript by presenting a general method for computing such probabilities and also detail a variety of directions for future research.