Parameterized mixed cluster editing via modular decomposition

In this paper we introduce a natural generalization of the well-known problems Cluster Editing and Bicluster Editing, whose parameterized versions have been intensively investigated in the recent literature. The generalized problem, called Mixed Cluster Editing or M-Cluster Editing, is formulated as follows. Let M be a family of graphs. Given a graph G and a nonnegative integer k, transform G, through a sequence of at most k edge editions, into a target graph G′ with the following property: G′ is a vertex-disjoint union of graphs G1, G2, . . . such that every Gi is a member of M. The graph G ′ is called a mixed cluster graph or M-cluster graph. Let K denote the family of complete graphs, Kl the family of complete l-partite graphs (l ≥ 2), and L = K ∪ Kl. In this work we focus on the case M = L. Using modular decomposition techniques previously applied to Cluster/Bicluster Editing, we present a linear-time algorithm to construct a problem kernel for the parameterized version of L-Cluster Editing.