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.