Abstract
The class of equistable graphs is defined by the existence of a cost structure on the vertices such that the maximal stable sets are characterized by their costs. This graph class, not contained in any nontrivial hereditary class, has so far been studied mostly from a structural point of view; characterizations and polynomial time recognition algorithms have been obtained for special cases.
We focus on complexity issues for equistable graphs and related classes. We describe a simple pseudo-polynomial-time dynamic programming algorithm to solve the maximum weight stable set problem along with the weighted independent domination problem in some classes of graphs, including equistable graphs. Our results are obtained within the wider context of Boolean optimization; corresponding hardness results are also provided. More specifically, we show that the above problems are APX-hard for equistable graphs and that it is co-NP-complete to determine whether a given cost function on the vertices of a graph defines an equistable cost structure of that graph.
Similar content being viewed by others
References
Brandstädt, A., Klembt, T., Lozin, V. V., & Mosca, R. (2008). On independent vertex sets in subclasses of apple-free graphs. In Lecture notes in computer science : Vol. 5369. Proceedings of ISAAC 2008 (pp. 849–859). Berlin: Springer.
Brandstädt, A., Le, V. B., & Mahfud, S. (2007). New applications of clique separator decomposition for the maximum weight stable set problem. Theoretical Computer Science, 70, 229–239.
Brandstädt, A., & Mahfud, S. (2002). Maximum weight stable set on graphs without claw and co-claw (and similar graph classes) can be solved in linear time. Information Processing Letters, 84, 251–259.
Erdős, P. (1955). Problems and results from additive number theory, Colloq. Théorie des nombres (pp. 127–137), Bruxells.
Garey, M. R., & Johnson, D. S. (1979). Computers and intractability. A guide to the theory of NP-completeness. New York: Freeman.
Giakoumakis, V., & Rusu, I. (1997). Weighted parameters in \((P_{5},\overline{P}_{5})\)-free graphs. Discrete Applied Mathematics, 80, 255–261.
Korach, E., & Peled, U. N. (2003). Equistable series-parallel graphs. Stability in graphs and related topics. Discrete Applied Mathematics, 132, 149–162.
Korach, E., Peled, U. N., & Rotics, U. (2008). Equistable distance-hereditary graphs. Discrete Applied Mathematics, 156, 462–477.
Lozin, V. V., & Milanič, M. (2008). A polynomial algorithm to find an independent set of maximum weight in a fork-free graph. Journal of Discrete Algorithms, 4, 595–604.
Mahadev, N. V. R., & Peled, U. N. (1995). Threshold graphs and related topics. Annals of discrete mathematics, 56. Amsterdam: North-Holland.
Mahadev, N. V. R., Peled, U. N., & Sun, F. (1994). Equistable graphs. J. Graph Theory, 18, 281–299.
Milanič, M., & Rudolf, G. (2009). Structural results for equistable graphs and related graph classes (RUTCOR Research Report 25-2009).
Mosca, R. (2008). Stable sets of maximum weight in (P 7, banner)-free graphs. Discrete Mathematics, 308, 20–33.
Payan, C. (1980). A class of threshold and domishold graphs: equistable and equidominating graphs. Discrete Mathematics, 29, 47–52.
Peled, U. N., & Rotics, U. (2003). Equistable chordal graphs. Stability in graphs and related topics. Discrete Applied Mathematics, 132, 203–210.
Rubin, A. (1981). A note on sum-distinct sets and a problem of Erdős. Abstracts of the AMS, 2, 42.
Shamir, A. (1979). On the cryptocomplexity of knapsack systems. In Proceedings 11th annual ACM symposium on theory of computing (pp. 118–129). New York: Association for Computing Machinery.
van Emde Boas, P. (1981). Another NP-complete partition problem and the complexity of computing short vectors in a lattice (Report No. 81-04). Department of Mathematics, University of Amsterdam.
Author information
Authors and Affiliations
Corresponding author
Additional information
Part of the work was done while M. Milanič was at the Universität Bielefeld. Support by the group “Combinatorial Search Algorithms in Bioinformatics” funded by the Sofja Kovalevskaja Award 2004 of the Alexander von Humboldt Stiftung and the German Federal Ministry of Research and Education is gratefully acknowledged.
Rights and permissions
About this article
Cite this article
Milanič, M., Orlin, J. & Rudolf, G. Complexity results for equistable graphs and related classes. Ann Oper Res 188, 359–370 (2011). https://doi.org/10.1007/s10479-010-0720-3
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10479-010-0720-3