We consider the two-player, complete information game of Cops and Robber played on undirected, finite, reflexive 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 \documentclass[12pt]{minimal}
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\begin{document}$${\tilde c(g)}$$\end{document} for non-orientable surfaces. It is known (Andreae, 1986) that, for a fixed surface, the maximum over all cop numbers of graphs embeddable in this surface is finite. More precisely, Quilliot (1985) showed that c(g) ≤ 2g + 3, and Schröder (2001) sharpened this to \documentclass[12pt]{minimal}
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\begin{document}$${c(g)\le \frac32g + 3}$$\end{document}. In his paper, Andreae gave the bound \documentclass[12pt]{minimal}
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\begin{document}$${\tilde c(g) \in O(g)}$$\end{document} with a weak constant, and posed the question whether a stronger bound can be obtained. Nowakowski & Schröder (1997) obtained \documentclass[12pt]{minimal}
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\begin{document}$${\tilde c(g) \le 2g+1}$$\end{document}. In this short note, we show \documentclass[12pt]{minimal}
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\begin{document}$${\tilde c(g) \leq c(g-1)}$$\end{document}, for any g ≥ 1. As a corollary, using Schröder’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, \documentclass[12pt]{minimal}
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\begin{document}$${\tilde c(3) \le 5}$$\end{document}, and \documentclass[12pt]{minimal}
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\begin{document}$${\tilde c(g) \le \frac32g + 3/2}$$\end{document} for all other g.
