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 $\L={ \cal K} \cup { \cal KL}$. 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.
Categories: cs.DS