Complexity of Secure Sets

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 $\text{co-}\mathrm{NP}$ -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. In this work, we pinpoint the complexity of this problem by showing that it is ${\Sigma }_2^P$ -complete. Furthermore, the problem has so far not been subject to a parameterized complexity analysis that considers structural parameters. In the present work, we prove that the problem is $W\left[1\right]$ -hard when parameterized by treewidth. This is surprising since the problem is known to be FPT when parameterized by solution size and "subset problems" that satisfy this property usually tend to be FPT for bounded treewidth as well. Finally, we give an upper bound by showing membership in $\mathrm{XP}$ , and we provide a positive result in the form of an FPT algorithm for checking whether a given set is secure on graphs of bounded treewidth.