Complexity of the cluster deletion problem on subclasses of chordal graphs

We consider the following vertex-partition problem on graphs, known as the CLUSTER DELETION (CD) problem: given a graph with real nonnegative edge weights, partition the vertices into clusters (in this case, cliques) to minimize the total weight of edges outside the clusters. The decision version of this optimization problem is known to be NP-complete even for unweighted graphs and has been studied extensively. We investigate the complexity of the decision CD problem for the family of chordal graphs, showing that it is NP-complete for weighted split graphs, weighted interval graphs and unweighted chordal graphs. We also prove that the problem is NP-complete for weighted cographs. Some polynomial-time solvable cases of the optimization problem are also identified, in particular CD for unweighted split graphs, unweighted proper interval graphs and weighted block graphs.