Abstract
In this paper we consider the following game: players must alternately color the lowest numbered uncolored vertex of a given graph G = ({1,2, ..., n}, E) with a color, taken from a given set C, such that never two adjacent vertices are colored with the same color. In one variant, the first player which is unable to move, loses the game. In another variant, player 1 wins the game if and only if the game ends with all vertices colored. We show that for both variants, the problem to determine whether there is a winning strategy for player 1 is PSPACE-complete for any C with |C| ≥ 3, but the problems are solvable in O(n + eα(e, n)), and O(n + e) time, respectively, if |C| = 2. We also give polynomial time algorithms for the problems with certain restrictions on the graphs and orderings of the vertices. We give some partial results for the versions, where no order for coloring the vertices is specified.
This research was partially supported by the ESPRIT II Basic Research Actions Program of the EC under contract no. 3075 (project ALCOM).
This is a preview of subscription content, log in via an institution to check access.
Preview
Unable to display preview. Download preview PDF.
References
A. Adachi, S. Iwata, and T. Kasai. Some combinatorial game problems require Ω(n k) time. J. ACM, 31:361–376, 1984.
H. L. Bodlaender. Classes of graphs with bounded treewidth. Technical Report RUU-CS-86-22, Dept. of Computer Science, Utrecht University, Utrecht, 1986.
H. L. Bodlaender. Improved self-reduction algorithms for graphs with bounded treewidth. Technical Report RUU-CS-88-29, Dept. of Computer Science, Utrecht University, 1988. To appear in: Proc. Workshop on Graph Theoretic Concepts in Comp. Sc. '89.
S. Even and R. Tarjan. A combinatorial problem which is complete in polynomial space. J. ACM, 23:710–719, 1976.
A. Fraenkel, M. Garey, D. Johnson, T. Schaefer, and Y. Yesha. The complexity of checkers on an n × n board — preliminary version. In Proc. 19th Annual Symp. on Foundations of Computer Science, pages 55–64, 1978.
A. Fraenkel and E. Goldschmidt. Pspace-hardness of some combinatorial games. J. Combinatorial Theory ser. A, 46:21–38, 1987.
A. Fraenkel and D. Lichtenstein. Computing a perfect strategy for n by n chess requires time exponential in n. J. Combinatorial Theory ser. A, 31:199–214, 1981.
M. R. Garey and D. S. Johnson. Computers and Intractability, A Guide to the Theory of NP-Completeness. W.H. Freeman and Company, New York, 1979.
D. Johnson. The NP-completeness column: An ongoing guide. J. Algorithms, 4:397–411, 1983.
Personal communication. 1990.
T. Lengauer. Black-white pebbles and graph separation. Acta Inf., 16:465–475, 1981.
D. Lichtenstein and M. Sipser. Go is polynomial-space hard. J. ACM, 27:393–401, 1980.
J. Robson. N by N Checkers is exptime complete. SIAM J. Comput., 13:252–267, 1984.
T. J. Schaefer. The complexity of satisfiability problems. In Proc. 10th Symp. on Theory of Computation, pages 216–226, 1978.
T. J. Schaefer. On the complexity of some two-person perfect-information games. J. Comp. Syst. Sc., 16:185–225, 1978.
L. J. Stockmeyer and A. K. Chandra. Provably difficult combinatorial games. SIAM J. Comput., 8:151–174, 1979.
R. E. Tarjan. Efficiency of a good but not linear set union algorithm. J. ACM, 22:215–225, 1975.
Author information
Authors and Affiliations
Editor information
Rights and permissions
Copyright information
© 1991 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Bodlaender, H.L. (1991). On the complexity of some coloring games. In: Möhring, R.H. (eds) Graph-Theoretic Concepts in Computer Science. WG 1990. Lecture Notes in Computer Science, vol 484. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-53832-1_29
Download citation
DOI: https://doi.org/10.1007/3-540-53832-1_29
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-53832-5
Online ISBN: 978-3-540-46310-8
eBook Packages: Springer Book Archive