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

In this paper, we study the computational complexity of \textsc{$s$-Club Cluster Vertex Deletion}. Given a graph, \textsc{$s$-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 \sloppy $\text{textsc}ClusterVertexDeletion(CVD)$. We provide a faster algorithm for \textsc{$s$-CVD} on \emph{interval graphs}. For each $s\ge 1$, we give an $O(n(n+m))$-time algorithm for \textsc{$s$-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 \etal (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 $\text{textsc}CVD$ on \emph{well-partitioned chordal graphs}, a graph class introduced by Ahn \etal ($\text{textsc}WG2020$) 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 $\text{textsc}WeightedBipartiteVertexCover$. This generalises a result of Cao \etal (Theor. Comput. Sci., 2018) on split graphs. We also show that for any even integer $s\ge 2$, \textsc{$s$-CVD} is NP-hard on well-partitioned chordal graphs.