Primal-Dual Cops and Robber

Cops and Robber is a family of two-player games played on graphs in which one player controls a number of cops and the other player controls a robber. In alternating turns, each player moves (all) their figures. The cops try to capture the robber while the latter tries to flee indefinitely. In this paper we consider a variant of the game played on a planar graph where the robber moves between adjacent vertices while the cops move between adjacent faces. The cops capture the robber if they occupy all incident faces. We prove that a constant number of cops suffices to capture the robber on any planar graph of maximum degree $\Delta$ if and only if $\Delta \leq 4$.