Cluster Editing with Vertex Splitting

Cluster Editing , also known as Correlation Clustering , is a well-studied graph modification problem. In this problem, one is given a graph and the task is to perform up to k edge additions or deletions to transform it into a cluster graph, i.e., a graph consisting of a disjoint union of cliques. In this paper, we introduce a variation of Cluster Editing we call Cluster Editing with Vertex Splitting that extends this model to settings where clusters may be overlapping. Specifically, we allow a new edit operation that divides a vertex into two new vertices, each with a subset of the original neighbors. This approach addresses the limitations of assuming disjoint clusters, while still inherently limiting the amount of overlap when the number of edits is small. We show that Cluster Editing with Vertex Splitting is NP-complete and fixed-parameter tractable when parameterized by the number of editing operations k . In particular, we obtain O ( 2 9 k log k + n + m ) -time algorithm and a 6 k -vertex kernel.