S-Club Cluster Vertex Deletion on Interval and Well-Partitioned Chordal Graphs

Abstract

AbstractIn this paper, we study the computational complexity of s -Club Cluster Vertex Deletion. Given a graph, s -Club Cluster Vertex Deletion (s -CVD) aims to delete the minimum number of vertices from the graph so that each connected component of the resulting graph has a diameter at most s. When \(s=1\), the corresponding problem is popularly known as Cluster Vertex Deletion (CVD). We provide a faster algorithm for s -CVD on interval graphs. For each \(s\ge 1\), we give an \(O(n(n+m))\)-time algorithm for s -CVD on interval graphs with n vertices and m edges. In the case of \(s=1\), our algorithm is a slight improvement over the \(O(n^3)\)-time algorithm of Cao et al. (Theor. Comput. Sci., 2018) and for \(s \ge 2\), it significantly improves the state-of-the-art running time \(\left( O\left( n^4\right) \right) \).We also give a polynomial-time algorithm to solve CVD on well-partitioned chordal graphs, a graph class introduced by Ahn et al. (WG 2020) as a tool for narrowing down complexity gaps for problems that are hard on chordal graphs, and easy on split graphs. Our algorithm relies on a characterisation of the optimal solution and on solving polynomially many instances of the Weighted Bipartite Vertex Cover. This generalises a result of Cao et al. (Theor. Comput. Sci., 2018) on split graphs. We also show that for any even integer \(s\ge 2\), s -CVD is NP-hard on well-partitioned chordal graphs.