Cops and robber on grids and tori

This paper is a contribution to the classical cops and robber problem on a graph, directed to two-dimensional grids and toroidal grids. These studies are generally aimed at determining the minimum number of cops needed to capture the robber and proposing algorithms for the capture. We apply some new concepts to propose a new solution to the problem on grids that was already solved under a different approach, and apply these concepts to give efficient algorithms for the capture on toroidal grids. As for grids, we show that two cops suffice even in a semi-torus (i.e. a grid with toroidal closure in one dimension) and three cops are necessary and sufficient in a torus. Then we treat the problem in function of any number k of cops, giving efficient algorithms for grids and tori and computing lower and upper bounds on the capture time. Conversely we determine the minimum value of k needed for any given capture time and study a possible speed-up phenomenon.