On the parameterized complexity of s-club cluster deletion problems

Abstract

We study the parameterized complexity of the s-Club Cluster Edge Deletion (s-Club Cluster Vertex Deletion) problem: Given a graph G and two integers s≥2 and k≥1, is it possible to remove at most k edges (vertices) from G such that each connected component of the resulting graph has diameter at most s? Both s-Club Cluster Edge Deletion and s-Club Cluster Vertex Deletion problems are known to be NP-hard already when s=2. We prove that they admit a fixed-parameter tractable algorithm when parameterized by s and the treewidth of the input graph. The proof is based on a unified algorithm that solves the more general problem in which both edges and vertices can be removed from the input graph to obtain a set of disjoint components with bounded diameter. Our approach can also be exploited to solve a related problem, namely s-Club Cover, which asks whether it is possible to cover the vertices of a graph with at most d different s-clubs, for some fixed d≥1 and s≥2.