  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.
