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.

Key Words: monopolies; lexicographic product graphs

2010 Mathematics Subject Classification: Primary 05C69,

Secondary 05C76