Graphs and Combinatorics

, Volume 30, Issue 1, pp 119–124 | Cite as

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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References

  1. 1.
    Aigner M., Fromme M.: A game of cops and robbers. Discrete Appl. Math. 8, 1–12 (1984)CrossRefMATHMathSciNetGoogle Scholar
  2. 2.
    Andreae T.: Note on a pursuit game played on graphs. Discrete Appl. Math. 9, 111–115 (1984)CrossRefMATHMathSciNetGoogle Scholar
  3. 3.
    Andreae T.: On a pursuit game played on graphs for which a minor is excluded. J. Combin. Th. Ser. B 41, 37–47 (1986)CrossRefMATHMathSciNetGoogle Scholar
  4. 4.
    Berarducci A., Intrigila B.: On the cop number of a graph. Adv. Appl. Math. 14, 389–403 (1993)CrossRefMATHMathSciNetGoogle Scholar
  5. 5.
    Cain, G.L.: Introduction to General Topology. Addison-Wesley, Boston (1994)Google Scholar
  6. 6.
    Clarke N.E., MacGillivray G.: Characterizations of k-copwin graphs. Discrete Math. 312, 1421–1425 (2012)CrossRefMATHMathSciNetGoogle Scholar
  7. 7.
    Mohar, B., Thomassen, C.: Graphs on Surfaces. Johns Hopkins University Press, Baltimore (2001)Google Scholar
  8. 8.
    Nowakowski, R.J., Schröder, B.S.W.: Bounding the cop number using the crosscap number (1997, Preprint)Google Scholar
  9. 9.
    Nowakowski R.J., Winkler P.: Vertex to vertex pursuit in a graph. Discrete Math. 43, 23–29 (1983)MathSciNetGoogle Scholar
  10. 10.
    Quilliot, A.: Jeux et Points Fixes sur les graphes. Thèse de 3ème cycle, Université de Paris VI, pp. 131–145 (1978)Google Scholar
  11. 11.
    Quilliot A.: A short note about pursuit games played on a graph with a given genus. J. Combin. Theory Ser. B 38, 89–92 (1985)CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    Schröder, B.S.W.: The copnumber of a graph is bounded by \({\lfloor\frac32\mathrm{genus}(G)\rfloor+3}\). In: “Categorical Perspectives”—Proceedings of the Conference in Honor of George Strecker’s 60th Birthday, pp. 243–263. Birkhäuser, Basel (2001)Google Scholar

Copyright information

© Springer Japan 2012

Authors and Affiliations

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

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