Graphs and Combinatorics

, Volume 30, Issue 1, pp 119–124

A Note on the Cops and Robber Game on Graphs Embedded in Non-Orientable Surfaces

  • Nancy E. Clarke
  • Samuel Fiorini
  • Gwenaël Joret
  • Dirk Oliver Theis
Original Paper

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.

Keywords

Games on graphs Cops and robber game Cop number Graphs on surfaces 

Mathematics subject Classification (2000)

05C99 05C10 91A43 

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Copyright information

© Springer Japan 2012

Authors and Affiliations

  • Nancy E. Clarke
    • 1
  • Samuel Fiorini
    • 1
  • Gwenaël Joret
    • 3
  • Dirk Oliver Theis
    • 4
  1. 1.Department of Mathematics and StatisticsAcadia UniversityWolfvilleCanada
  2. 2.Département de MathématiqueUniversité Libre de BruxellesBrusselsBelgium
  3. 3.Département d’InformatiqueUniversité Libre de BruxellesBrusselsBelgium
  4. 4.Fakultät für MathematikOtto-von-Guericke-Universität MagdeburgMagdeburgGermany

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