The cluster deletion problem for cographs

The min-edge clique partition problem asks to find a partition of the vertices of a graph into a set of cliques with the fewest edges between cliques. This is a known NP-complete problem and has been studied extensively in the scope of fixed-parameter tractability (PT) 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. We prove new structural properties of cographs which characterize how a largest clique interacts with the rest of the graph. These results imply a remarkably simple polynomial time algorithm for CLUSTER DELETION on cographs. In contrast, we observe that CLUSTER DELETION remains NP-hard on a hereditary graph class which is slightly larger than cographs. Crown Copyright (C) 2013 Published by Elsevier B.V. All rights reserved.