We consider a simple graph
without isolated vertices and of minimum degree
.
Let
be an integer number such that
.
A vertex
of
is said to be
-controlled by a set
, if
where
represents the number of neighbors
has in
and
the degree
of
. The set
is called a
-monopoly if it
-controls every vertex
of
. The minimum cardinality of any
-monopoly in
is the
-monopoly number of
. In this article we study the
-monopolies of
the lexicographic product of graphs. Specifically we obtain several
relationships between the
-monopoly number of this product graph and the
-monopoly numbers and/or order of its factors. Moreover, we bound (or
compute the exact value) of the
-monopoly number of several families of
lexicographic product graphs.