A Little Note on the Cops & Robber Game on Graphs Embedded in Non-orientable Surfaces

The two-player, complete information game of Cops and Robber is played on undirected finite graphs. A number of cops and one robber are positioned on vertices and take turns in sliding along edges. The cops win if, after a move, a cop and the robber are on the same vertex. The minimum number of cops needed to catch the robber on a graph is called the cop number of that graph. Let c(g) be the supremum over all cop numbers of graphs embeddable in a closed orientable surface of genus g, and likewise c̃(g) for non-orientable surfaces. It is known (Andrea, 1986) that if X is a fixed surface, the maximum over all cop numbers of graphs embeddable in this surface is finite. More precisely, Quilliot (1983 probably) showed that c(g) ≤ 2g + O(1), and Schroeder (2001) sharpened this to c(g) ≤ 3 2 g + 3. In his paper, Andrea gave the bound c̃(g) ≤ O(n) with a weak constant, and posed the question whether a stronger bound can be obtained. In a recent preprint, Nowakowski & Schroeder obtained c̃op(g) ≤ 2g + 1. In this short note, we show c̃(g) = c(g − 1) for any g ≥ 1. As a corollary, using Schroeder’s results, we obtain the following: the maximum cop number of graphs embeddable in the projective plane is 3; the maximum cop number of graphs embeddable in the Klein Bottle is at most 4, c̃(3) ≤ 5, and c̃(g) ≤