On the Complexity of 2-Club Cluster Editing with Vertex Splitting
摘要
Editing a graph to obtain a disjoint union of s-clubs is one of the models for correlation clustering, which seeks a partition of the vertex set of a graph so that elements of each resulting set are close enough according to some given criterion. For example, in the case of editing into s-clubs, the criterion is proximity since any pair of vertices (in an s-club) is within a distance of s from each other. In this work, we consider the vertex splitting operation, which allows a vertex to belong to more than one cluster. This operation was studied as one of the parameters associated with the Cluster Editing problem. We study the complexity and parameterized complexity of the 2-Club Cluster Edge Deletion with Vertex Splitting and 2-Club Cluster Vertex Splitting problems. We prove that both problems are \(\textsf {NP}\) -complete and \(\textsf {APX}\) -hard. On the positive side, we show that the two problems are solvable in polynomial-time on trees and that they are both fixed-parameter tractable with respect to the number of allowed editing operations.