PIMS-UVic Discrete Math Seminar: Vesna Iršič

In 2021, Mohar introduced the game of Cops and Robber on geodesic spaces. The game captures the behavior of the Cops and Robber game played on graphs and that of continuous pursuit-evasion games. Analogous to one of the main open problems for the Cops and Robber game on graphs, Mohar conjectured that the cop number of a geodesic surface of genus $g$ is at most $O(\sqrt{g})$. Surprisingly, this upper bound can be significantly improved on surfaces of constant curvature which will be the main focus of this talk.

It turns out that the cop number of compact spherical and Euclidean surfaces is at most $2$. Even more surprisingly, the cop number of compact hyperbolic surfaces is also at most $2$, independently of their genus. We will also consider the strong cop number of these surfaces and present several generalizations to higher-dimensions.

Joint work with Bojan Mohar and Alexandra Wesolek.