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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)

Abstract

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.

Keywords

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

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References

  1. 1.
    Fomin, F., Golovach, P., Hall, A., Mihalák, M., Vicari, E., Widmayer, P.: How to Guard a Graph? Algorithmica, doi:10.1007/s00453-009-9382-4Google Scholar
  2. 2.
    Stockmeyer, L., Chandra, A.: Provably Difficult Combinatorial Games. SIAM J. Comput. 8(2), 151–174 (1979)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Book, R.V.: Comparing complexity classes. J. of Computer and System Sciences 9(2), 213–229 (1974)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Alspach, B.: Searching and sweeping graphs: a brief survey. Matematiche (Catania) 59(1-2), 5–37 (2006)MathSciNetzbMATHGoogle Scholar
  5. 5.
    Nowakowski, R., Winkler, P.: Vertex-to-vertex pursuit in a graph. Discrete Math. 43(2-3), 235–239 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Quilliot, A.: Some results about pursuit games on metric spaces obtained through graph theory techniques. European J. Combin. 7(1), 55–66 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Aigner, M., Fromme, M.: A game of cops and robbers. Discrete Appl. Math. 8(1), 1–11 (1984)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Nagamochi, H.: Cop-robber guarding game with cycle robber-region. Theoretical Computer Science 412, 383–390 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Thirumala Reddy, T.V., Sai Krishna, D., Pandu Rangan, C.: The Guarding Problem – Complexity and Approximation. In: Fiala, J., Kratochvíl, J., Miller, M. (eds.) IWOCA 2009. LNCS, vol. 5874, pp. 460–470. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  10. 10.
    Fomin, F.V., Thilikos, D.M.: An annotated bibliography on guaranteed graph searching. Theor. Comp. Sci. 399, 236–245 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Goldstein, A.S., Reingold, E.M.: The complexity of pursuit on a graph. Theor. Comp. Sci. 143, 93–112 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Fomin, F.V., Golovach, P.A., Kratochvíl, J., Nisse, N., Suchan, K.: Pursuing a fast robber on a graph. Theor. Comp. Sci. 411, 1167–1181 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Fomin, F.V., Golovach, P.A., Lokshtanov, D.: Guard games on graphs: Keep the intruder out! In: Bampis, E., Jansen, K. (eds.) WAOA 2009. LNCS, vol. 5893, pp. 147–158. Springer, Heidelberg (2010)CrossRefGoogle Scholar
  14. 14.
    Fomin, F.V., Golovach, P.A., Kratochvíl, J.: On tractability Cops and Robbers Game. In: Proceedings of the 5th IFIP International Conference on Theoretical Computer Science (TCS 2008). IFIP, vol. 237, pp. 171–185. Springer, Heidelberg (2008)CrossRefGoogle Scholar
  15. 15.
    Anderson, M., Barrientos, C., Brigham, R., Carrington, J., Vitray, R., Yellen, J.: Maximum demand graphs for eternal security. J. Combin. Math. Combin. Comput. 61, 111–128 (2007)MathSciNetzbMATHGoogle Scholar
  16. 16.
    Burger, A.P., Cockayne, E.J., Grundlingh, W.R., Mynhardt, C.M., van Vuuren, J.H., Winterbach, W.: Infinite order domination in graphs. J. Combin. Math. Combin. Comput. 50, 179–194 (2004)MathSciNetzbMATHGoogle Scholar
  17. 17.
    Goddard, W., Hedetniemi, S.M., Hedetniemi, S.T.: Eternal security in graphs. J. Combin. Math. Combin. Comput. 52, 169–180 (2005)MathSciNetzbMATHGoogle Scholar
  18. 18.
    Goldwasser, J., Klostermeyer, W.F.: Tight bounds for eternal dominating sets in graphs. Discrete Math. 308, 2589–2593 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Klostermeyer, W.F.: Complexity of Eternal Security. J. Comb. Math. Comb. Comput. 61, 135–141 (2007)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Klostermeyer, W.F., MacGillivray, G.: Eternal security in graphs of fixed independence number. J. Combin. Math. Combin. Comput. 63, 97–101 (2007)MathSciNetzbMATHGoogle Scholar
  21. 21.
    Klostermeyer, W.F., MacGillivray, G.: Eternally Secure Sets, Independence Sets, and Cliques. AKCE International Journal of Graphs and Combinatorics 2, 119–122 (2005)MathSciNetzbMATHGoogle Scholar
  22. 22.
    Klostermeyer, W.F., MacGillivray, G.: Eternal dominating sets in graphs. J. Combin. Math. Combin. Comput. 68, 97–111 (2009)MathSciNetzbMATHGoogle Scholar

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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