Complexity of the Cop and Robber Guarding Game

  • Robert Šámal
  • Rudolf Stolař
  • Tomas Valla
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7056)


The guarding game is a game in which several cops has to guard a region in a (directed or undirected) graph against a robber. The robber and the cops are placed on vertices of the graph; they take turns in moving to adjacent vertices (or staying), cops inside the guarded region, the robber on the remaining vertices (the robber-region). The goal of the robber is to enter the guarded region at a vertex with no cop on it. The problem is to determine whether for a given graph and given number of cops the cops are able to prevent the robber from entering the guarded region. The problem is highly nontrivial even for very simple graphs. It is known that when the robber-region is a tree, the problem is NP-complete, and if the robber-region is a directed acyclic graph, the problem becomes PSPACE-complete [Fomin, Golovach, Hall, Mihalák, Vicari, Widmayer: How to Guard a Graph? Algorithmica, DOI: 10.1007/s00453-009-9382-4]. We solve the question asked by Fomin et al. in the previously mentioned paper and we show that if the graph is arbitrary (directed or undirected), the problem becomes E-complete.


pursuit game cops and robber game graph guarding game computational complexity E-completeness 


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Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Robert Šámal
    • 1
  • Rudolf Stolař
    • 1
  • Tomas Valla
    • 1
  1. 1.Faculty of Mathematics and Physics, Institute for Theoretical Computer Science (ITI)Charles UniversityPragueCzech Republic

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