s-Club Cluster Vertex Deletion on interval and well-partitioned chordal graphs

In 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 ≥ 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 (n³)-time algorithm of Cao et al. (2018), and for s ≥ 2, it significantly improves the state-of-the-art running time (O (n⁴)).

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. (2018) on split graphs. We also show that for any even integer s ≥ 2, s-CVD is NP-hard on well-partitioned chordal graphs.