Complexity of the cluster deletion problem on chordal graphs, subclasses of chordal graphs, and cographs∗

We consider the following vertex-partition problem on graphs: given a graph with real nonnegative edge weights, partition the vertices into clusters (i.e. cliques) to minimize the total weight of edges out of the clusters. This optimization problem is known to be an NP-complete problem even for unweighted graphs and has been studied extensively in the scope of fixed-parameter tractability (FPT), where it is commonly known as the CLUSTER DELETION problem. Many of the recently-developed FPT algorithms rely on being able to solve CLUSTER DELETION in polynomial-time on restricted graph structures. In this paper, the complexity of the CLUSTER DELETION problem is investigated for the family of chordal graphs. It is shown that this problem is NP-complete for edge-weighted split graphs, edge-weighted interval graphs and edge-unweighted chordal graphs. Some polynomial-time solvable cases are also identified, in particular CLUSTER DELETION for unweighted split graphs, unweighted proper interval graphs and weighted block graphs.