Abstract
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 \({\tilde c(g)}\) 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 \({c(g)\le \frac32g + 3}\). In his paper, Andreae gave the bound \({\tilde c(g) \in O(g)}\) with a weak constant, and posed the question whether a stronger bound can be obtained. Nowakowski & Schröder (1997) obtained \({\tilde c(g) \le 2g+1}\). In this short note, we show \({\tilde c(g) \leq c(g-1)}\), 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, \({\tilde c(3) \le 5}\), and \({\tilde c(g) \le \frac32g + 3/2}\) for all other g.
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Samuel Fiorini: This work was supported in part by the Actions de Recherche Concertées (ARC) fund of the Communauté Française de Belgique. Gwenaël Joret: Postdoctoral Researcher of the Fonds National de la Recherche Scientifique (F.R.S.–FNRS).
Dirk Oliver Theis: Supported by Fonds National de la Recherche Scientifique (F.R.S.–FNRS).
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Clarke, N.E., Fiorini, S., Joret, G. et al. A Note on the Cops and Robber Game on Graphs Embedded in Non-Orientable Surfaces. Graphs and Combinatorics 30, 119–124 (2014). https://doi.org/10.1007/s00373-012-1246-z
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DOI: https://doi.org/10.1007/s00373-012-1246-z