Jumping robbers in digraphs

Abstract

In cops and robber games, a team of cops tries to capture a robber on a given graph. The cops occupy some vertices of the graph and the robber occupies one vertex. Given concrete rules how the players can move and when the robber is captured, the minimum number of cops needed to capture the robber is a natural parameter of the graph.Richerby and Thilikos [25] generalise the game to a game where the cops must capture multiple robbers. They show that the number of cops needed to capture r robbers is at most k⋅log⁡r where k is the minimum number of cops who can capture one robber. Furthermore, this bound is tight. Moreover, capturing every new robber may demand some new cops and infinitely many robbers demand as many cops as one invisible robber.We generalise the results of Richerby and Thilikos in two aspects. We give a linear bound on the number of additional cops in the game (1) on directed graphs and (2) where the robbers have more power: every robber can leave his vertex and jump to another robber. We show that our bound is tight and that the reason for a worse lower bound is not the directedness of the graphs, but the ability of the robbers to jump. We also prove that adding new robbers induces a hierarchy of the cop numbers for a fixed graph which is strict on some graphs and converges to the minimum number of cops needed to capture an invisible robber (as in [25]).