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 On4. 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.

A graph $G$ is said to be $k$-distinguishable if the vertex set can be colored using $k$ colors such that no non-trivial automorphism fixes every color class, and the distinguishing number $D(G)$ is the least integer $k$ for which $G$ is $k$-distinguishable. If for each $v\in V(G)$ we have a list $L(v)$ of colors, and we stipulate that the color assigned to vertex $v$ comes from its list $L(v)$ then $G$ is said to be $\mathcal{L}$-distinguishable where $\mathcal{L} =\{L(v)\}_{v\in V(G)}$. The list distinguishing number of a graph, denoted $D_l(G)$, is the minimum integer $k$ such that every collection of lists $\mathcal{L}$ with $|L(v)|=k$ admits an $\mathcal{L}$-distinguishing coloring. In this paper, we prove that $D_l(G)=D(G)$ when $G$ is a Kneser graph.