Catching a Fast Robber on Interval Graphs

  • Tomáš Gavenčiak
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6648)


We analyse the Cops and ∞-fast Robber game on the class of interval graphs and show it to be polynomially decidable on such graphs. This solves an open problem posed in paper “Pursuing a fast robber on a graph” by Fomin et al. [4] The game is known to be already NP-hard on chordal graphs and split-graphs.

The game is played by two players, one controlling k cops, the other a robber. The players alternate in turns, all the cops move at once to distance at most one, the robber moves along any cop-free path. Cops win by capturing the robber, the robber by avoiding capture.

The analysis relies on the properties of an interval representation of the graph. We show that while the game-state graph is generally exponential, every cops’ move can be decomposed into simple moves of three types, and the states are reduced to those defined by certain cuts of the interval representation. This gives a restricted game equivalent to the original one together with a winning strategy computable in polynomial time.


cop and robber game pursuit game combinatorial game interval graph interval graph representation 


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  1. 1.
    Albert, M.H., Nowakowski, R.J., Wolfe, D.: Lessons in Play: An Introduction to Combinatorial Game Theory. AK Peters, USA (2007)zbMATHGoogle Scholar
  2. 2.
    Bollobás, B.: Modern graph theory. Graduate Texts in Mathematics, vol. 184. Springer, New York (1998)zbMATHGoogle Scholar
  3. 3.
    Brandstädt, A.: The computational complexity of feedback vertex set, hamiltonian circuit, dominating set, steiner tree, and bandwidth on special perfect graphs. Elektronische Informationsverarbeitung und Kybernetik 23(8/9), 471–477 (1987)MathSciNetzbMATHGoogle Scholar
  4. 4.
    Fomin, F.V., Golovach, P., Kratochvil, J., Nisse, N., Suchan, K.: Pursuing a fast robber on a graph. Theoretical Computer Science (2009) (submitted)Google Scholar
  5. 5.
    Fomin, F.V., Golovach, P.A., Kratochvíl, J.: On tractability of cops and robbers game. In: IFIP TCS, pp. 171–185 (2008)Google Scholar
  6. 6.
    Korte, N., Möhring, R.H.: An incremental linear-time algorithm for recognizing interval graphs. SIAM J. Comput. 18(1), 68–81 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Nowakowski, R., Winkler, P.: Vertex to vertex pursuit in a graph. Discrete Math. 43(2), 235–239 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Quilliot, A.: A short note about pursuit games played on a graph with a given genus. Journal of Combinatorial Theory, Series B 38(1), 89–92 (1985)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Tomáš Gavenčiak
    • 1
  1. 1.Department of Applied MathematicsCharles UniversityPragueCzech Republic

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