Kayles on special classes of graphs — An application of Sprague-Grundy theory

  • Hans L. Bodlaender
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 657)


Kayles is the game, where two players alternately choose a vertex that has not been chosen before nor is adjacent to an already chosen vertex from a given graph. The last player that choses a vertex wins the game. We show, with help of Sprague-Grundy theory, that the problem to determine which player has a winning strategy for a given graph, can be solved in O(n3 time on interval graphs, on circular arc graphs, on permutation graphs, and on co-comparability graphs and in O(n1.631) time on cographs. For general graphs, the problem is known to be PSPACE-complete, but can be solved in time, polynomial in the number of isolatable sets of vertices of the graph.


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  1. 1.
    E. R. Berlekamp, J. H. Conway, and R. K. Guy. Winning Ways for your mathematical plays, Volume 1: Games in General. Academic Press, New York, 1982.Google Scholar
  2. 2.
    K. S. Booth and G. S. Lueker. Testing for the consecutive ones property, interval graphs, and graph planarity using pq-tree algorithms. J. Comp. Syst. Sc., 13:335–379, 1976.Google Scholar
  3. 3.
    J. H. Conway. On Numbers and Games. Academic Press, London, 1976.Google Scholar
  4. 4.
    D. G. Corneil, Y. Perl, and L. K. Stewart. A linear recognition algorithm for cographs. SIAM J. Comput., 4:926–934, 1985.Google Scholar
  5. 5.
    M. Dietzfelbinger, A. Karlin, K. Mehlhorn, F. Meyer auf der Heide, H. Rohnert, and R. E. Tarjan. Dynamic perfect hashing: Upper and lower bounds. In Proceedings of the 29th Annual Symposium on Foundations of Computer Science, pages 524–531, 1988.Google Scholar
  6. 6.
    M. C. Golumbic. Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York, 1980.Google Scholar
  7. 7.
    M. C. Golumbic, D. Rotem, and J. Urrutia. Comparability graphs and intersection graphs. Disc. Math., 43:37–46, 1983.Google Scholar
  8. 8.
    D. König. Theorie der endlichen und unendlichen Graphen. Akademische Verlagsgesellschaft, Leipzig, 1936.Google Scholar
  9. 9.
    T. J. Schaefer. On the complexity of some two-person perfect-information games. J. Comp. Syst. Sc., 16:185–225, 1978.Google Scholar
  10. 10.
    A. Tucker. An efficient test for circular-arc graphs. SIAM J. Comput., 9:1–24, 1980.Google Scholar
  11. 11.
    P. van Emde Boas, R. Kaas, and E. Zijlstra. Design and implementation of an efficient priority queue. Mathematical Systems Theory, 10:99–127, 1984.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1993

Authors and Affiliations

  • Hans L. Bodlaender
    • 1
  1. 1.Department of Computer ScienceUtrecht UniversityTB UtrechtThe Netherlands

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