Abstract
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 (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.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.
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