Complexity of Secure Sets

TL;DR

This paper investigates the computational complexity of secure sets in graphs, establishing its hardness in various parameterized contexts and providing algorithms for specific cases.

Contribution

It proves the problem is -complete, -hard for treewidth, and offers an FPT algorithm for graphs with bounded treewidth.

Findings

Deciding secure sets is -complete.

The problem is -hard when parameterized by treewidth.

An FPT algorithm exists for graphs with bounded treewidth.

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 co-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 $\Sigma^P_2$-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 $W[1]$-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 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.