Complexity of Secure Sets

Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9224)

Abstract

A secure set S in a graph is defined as a set of vertices such that for any \(X\subseteq S\) the majority of vertices in the neighborhood of X belongs to S. It is known that deciding whether a set S is secure in a graph is \(\text {co-}\hbox {NP}\)-complete. However, it is still open how this result contributes to the actual complexity of deciding whether, for a given graph G and integer k, a non-empty secure set for G of size at most k exists. While membership in the class \(\Sigma ^{\mathrm{P}}_{2}\) is rather easy to see for this existence problem, showing \(\Sigma ^{\mathrm{P}}_{2}\)-hardness is quite involved. In this paper, we provide such a hardness result, hence classifying the secure set existence problem as \(\Sigma ^{\mathrm{P}}_{2}\)-complete. We do so by first showing hardness for a variantof the problem, which we then reduce step-by-step to secure set existence. In total, we obtain eight new completeness results for different variants of the secure set existence problem.

Keywords

Computational complexity Complexity analysis Secure sets 

Notes

Acknowledgments

This work was supported by the Austrian Science Fund (FWF) projects P25607 and Y698. We would like to thank the reviewers for their valuable comments and Herbert Fleischner for drawing our attention to the problem of secure sets.

References

  1. 1.
    Abseher, M., Bliem, B., Charwat, G., Dusberger, F., Woltran, S.: Computing secure sets in graphs using answer set programming. In: Proceedings of ASPOCP 2014 (2014)Google Scholar
  2. 2.
    Brewka, G., Eiter, T., Truszczyński, M.: Answer set programming at a glance. Commun. ACM 54(12), 92–103 (2011)CrossRefGoogle Scholar
  3. 3.
    Brigham, R.C., Dutton, R.D., Hedetniemi, S.T.: Security in graphs. Discrete Appl. Math. 155(13), 1708–1714 (2007)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    de Haan, R., Szeider, S.: The parameterized complexity of reasoning problems beyond NP. In: Proceedings of KR 2014, pp. 82–91 (2014)Google Scholar
  5. 5.
    Enciso, R.I., Dutton, R.D.: Parameterized complexity of secure sets. Congr. Numer. 189, 161–168 (2008)MathSciNetMATHGoogle Scholar
  6. 6.
    Ho, Y.Y.: Global secure sets of trees and grid-like graphs. Ph.D. thesis, University of Central Florida, Orlando, USA (2011)Google Scholar
  7. 7.
    Marx, D.: Complexity of clique coloring and related problems. Theor. Comput. Sci. 412(29), 3487–3500 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Szeider, S.: Generalizations of matched CNF formulas. Ann. Math. Artif. Intell. 43(1), 223–238 (2005)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2016

Authors and Affiliations

  1. 1.Institute of Information Systems 184/2TU WienViennaAustria

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