Pursuing a fast robber on a graph

  • Authors:

  • Fedor V. Fomin;Petr A. Golovach;Jan Kratochvíl;Nicolas Nisse;Karol Suchan

  • Affiliations:

  • Department of Informatics, University of Bergen, N-5020 Bergen, Norway;Department of Informatics, University of Bergen, N-5020 Bergen, Norway;Institute for Theoretical Computer Science, Department of Applied Mathematics, Charles University, Praha, Czech Republic;MASCOTTE, INRIA, I3S, CNRS, UNS, Sophia Antipolis, France;Facultad de Ingeniería y Ciencias, Universidad Adolfo Ibáñez, Santiago, Chile and Faculty of Applied Mathematics, AGH - University of Science and Technology, Cracow, Poland

  • Venue:
  • Theoretical Computer Science
  • Year:
  • 2010

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Abstract

The Cops and Robbers game as originally defined independently by Quilliot and by Nowakowski and Winkler in the 1980s has been much studied, but very few results pertain to the algorithmic and complexity aspects of it. In this paper we prove that computing the minimum number of cops that are guaranteed to catch a robber on a given graph is NP-hard and that the parameterized version of the problem is W[2]-hard; the proof extends to the case where the robber moves s time faster than the cops. We show that on split graphs, the problem is polynomially solvable if s=1 but is NP-hard if s=2. We further prove that on graphs of bounded cliquewidth the problem is polynomially solvable for s@?2. Finally, we show that for planar graphs the minimum number of cops is unbounded if the robber is faster than the cops.