On the Parameterized Complexity of s-club Cluster Deletion Problems

We study the parameterized complexity of the $$s$$ -Club Cluster Edge Deletion problem: Given a graph G and two integers $$s \ge 2$$ and $$k \ge 1$$ , is it possible to remove at most k edges from G such that each connected component of the resulting graph has diameter at most s? This problem is known to be $$\texttt{NP}$$ -hard already when $$s = 2$$ .We prove that it admits a fixed-parameter tractable algorithm when parameterized by s and the treewidth of the input graph. The same result easily transfers to the case in which we can remove at most k vertices, rather than k edges, from G such that each connected component of the resulting graph has diameter at most s, namely to $$s$$ -Club Cluster Vertex Deletion.