# Complexity of Secure Sets

## 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.

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