On Structural Parameterizations of the Harmless Set Problem

In this paper, we study the Harmless Set problem from a parameterized complexity perspective. Given a graph \(G = (V,E)\), a threshold function\(~t~:~ V \rightarrow {\mathbb {N}}\) and an integer k, we study Harmless Set, where the goal is to find a subset of vertices \(S \subseteq V\) of size at least k such that every vertex \(v\in V\) has fewer than t(v) neighbours in S. On the positive side, we obtain fixed-parameter algorithms for the problem when parameterized by the neighbourhood diversity, the twin-cover number and the vertex integrity of the input graph. We complement two of these results from the negative side. On dense graphs, we show that the problem is W[1]-hard parameterized by cluster vertex deletion number—a natural generalization of the twin-cover number. We show that the problem is W[1]-hard parameterized by a wide range of fairly restrictive structural parameters such as the feedback vertex set number, pathwidth, and treedepth—a natural generalization of the vertex integrity. We thereby resolve one open question stated by Bazgan and Chopin (Discrete Optim 14(C):170–182, 2014) concerning the complexity of Harmless Set parameterized by the treewidth of the input graph. We also show that Harmless Set for a special case where each vertex has the threshold set to half of its degree (the so-called Majority Harmless Set problem) is W[1]-hard when parameterized by the treewidth of the input graph. Given a graph G and an irredundant c-expression of G, we prove that Harmless Set can be solved in XP-time when parameterized by clique-width.