On Necessary and Sufficient Number of Cops in the Game of Cops and Robber in Multidimensional Grids

We theoretically analyze the Cops and Robber Game for the first time in a multidimensional grid. It is shown that for an $n$-dimensional grid, at least $n$ cops are necessary to ensure capture of the robber. We also present a set of cop strategies for which $n$ cops are provably sufficient to catch the robber. Further, for two-dimensional grid, we provide an efficient cop strategy for which the robber is caught even by a single cop under certain conditions.Comment: This is a revised and extended version of the poster paper with the same title that has been presented in the 8th Asian Symposium on Computer Mathematics (ASCM), December 15-17, 2007, Singapor

A probabilistic version of the game of Zombies and Survivors on graphs

We consider a new probabilistic graph searching game played on graphs, inspired by the familiar game of Cops and Robbers. In Zombies and Survivors, a set of zombies attempts to eat a lone survivor loose on a given graph. The zombies randomly choose their initial location, and during the course of the game, move directly toward the survivor. At each round, they move to the neighbouring vertex that minimizes the distance to the survivor; if there is more than one such vertex, then they choose one uniformly at random. The survivor attempts to escape from the zombies by moving to a neighbouring vertex or staying on his current vertex. The zombies win if eventually one of them eats the survivor by landing on their vertex; otherwise, the survivor wins. The zombie number of a graph is the minimum number of zombies needed to play such that the probability that they win is strictly greater than 1/2. We present asymptotic results for the zombie numbers of several graph families, such as cycles, hypercubes, incidence graphs of projective planes, and Cartesian and toroidal grids

Statistical Model Checking for Cops and Robbers Game on Random Graph Models

Cops and robbers problem has been studied over the decades with many variants and applications in graph searching problem. In this work, we study a variant of cops and robbers problem on graphs. In this variant, there are di�erent types of cops and a minimum number of each type of cops are required to catch a robber. We studied this model over various random graph models and analyzed the properties using statistical model checking. To the best of our knowledge this variant of the cops and robber problem has not been studied yet. We have used statistical techniques to estimate the probability of robber getting caught in di�erent random graph models. We seek to compare the ease of catching robbers performing random walk on graphs, especially complex networks. In this work, we report the experiments that yields interesting empirical results. Through the experiments we have observed that it is easier to catch a robber in Barab�asi Albert model than in Erd�os-R�enyi graph model. We have also experimented with k-Regular graphs and real street networks. In our work, the model is framed as the multi-agent based system and we have implemented a statistical model checker, SMCA tool which veri�es agents based systems using statistical techniques. SMCA tool can take the model in JAVA programming language and support Probabilistic - Bounded LTL logic for property specification