Cops and robbers on string graphs

Abstract

The game of cops and robber, introduced by Nowakowski and Winkler in 1983, is played by two players on a graph. One controls k cops and the other a robber. The players alternate and move their pieces to the distance at most one. The cops win if they capture the robber, the robber wins by escaping indefinitely. The cop number of G is the smallest k such that k cops win the game. We extend the results of Gavenčiak et al. [ISAAC 2013], investigating the maximum cop number of geometric intersection graphs. Our main result shows that the maximum cop number of string graphs is at most 15, improving the previous bound 30. We generalize this approach to string graphs on a surface of genus g to show that the maximum cop number is at most 10g + 15, which strengthens the result of Quilliot [J. Combin. Theory Ser. B 38, 89–92 (1985)]. For outer string graphs, we show that the maximum cop number is between 3 and 4. Our results also imply polynomial-time algorithms determining the cop number for all these graph classes.