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# 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

## 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)
3. 3.
Brigham, R.C., Dutton, R.D., Hedetniemi, S.T.: Security in graphs. Discrete Appl. Math. 155(13), 1708–1714 (2007)
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)
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)
8. 8.
Szeider, S.: Generalizations of matched CNF formulas. Ann. Math. Artif. Intell. 43(1), 223–238 (2005)

## Copyright information

© Springer-Verlag Berlin Heidelberg 2016

## Authors and Affiliations

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