A secure set S in a graph is defined as a set of vertices such that for any $$X\subseteq S$$X⊆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 $$\mathrm {\text {co-}NP}$$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 $$\mathrm {\Sigma ^P_2}$$Σ2P-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 $$\mathrm {W[1]}$$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 $$\mathrm {XP}$$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.

中文翻译：

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安全集的复杂性

图中的安全集合 S 被定义为一组顶点，使得对于任何 $$X\subseteq S$$X⊆S，X 邻域中的大多数顶点都属于 S。 S 在图中是安全的，是 $$\mathrm {\text {co-}NP}$$co-NP-complete。然而，这个结果如何影响决定对于给定的图 G 和整数 k 是否存在最大大小为 k 的 G 的非空安全集的实际复杂性仍然是开放的。在这项工作中，我们通过证明它是 $$\mathrm {\Sigma ^P_2}$$Σ2P-complete 来确定这个问题的复杂性。此外，迄今为止，该问题尚未进行考虑结构参数的参数化复杂性分析。在目前的工作中，我们证明了当由树宽参数化时问题是 $$\mathrm {W[1]}$$W[1]-hard。这是令人惊讶的，因为当通过解决方案大小和“子集问题”进行参数化时，已知问题是 FPT，而满足此属性的“子集问题”通常也倾向于是有界树宽的 FPT。最后，我们通过显示 $$\mathrm {XP}$$XP 中的成员资格给出了一个上限，并且我们以 FPT 算法的形式提供了一个积极的结果，用于检查给定集合在有界树宽的图上是否安全。