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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 ${\cal M}$-Cluster Editing, is formulated as follows. Let ${\cal M}$ 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 $G_1, G_2, \ldots$ such that every $G_i$ is a member of ${\cal M}$. The graph $G'$ is called a mixed cluster graph or ${\cal M}$-cluster graph. Let ${\cal K}$ denote the family of complete graphs, ${\cal KL}$ the family of complete $l$-partite graphs ($l \geq 2$), and $Ł={\cal K} \cup {\cal KL}$. In this work we focus on the case ${\cal M} = {\cal L}$. 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 ${\cal L}$-Cluster Editing. Keywords: bicluster graphs, cluster graphs, edge edition problems, edge modification problems, fixed-parameter tractability, NP-complete problems.