  Let $G$ be any simple and undirected graph. By a threshold assignment $\tau$
in $G$ we mean any function $\tau:V(G)\rightarrow \mathbb{N}$ such that
$\tau(v)\leq d_G(v)$ for any vertex $v$ of $G$. Given a graph $G$ with a
threshold assignment $\tau$, a subset of vertices $M$ is said to be a
$\tau$-monopoly if there exist at least $\tau(v)$ neighbors in $M$ for any
vertex $v \in V(G) \setminus M$. Similarly, a subset of vertices $D$ is said to
be a $\tau$-dynamic monopoly if starting with the set $D$ and iteratively
adding to the current set further vertices $u$ that have at least $\tau(u)$
neighbors in it, results in the entire vertex set $V(G)$. Denote by
$mon_{\tau}(G)$ (resp. $dyn_{\tau}(G)$) the smallest cardinality of a
$\tau$-monopoly (resp. $\tau$-dynamic monopoly) of the graph among all others.
In this paper we obtain some lower and upper bounds for these two parameters
with constant threshold assignments for Cartesian product graphs. Our bounds
improve the previous known bounds. We also determine the exact value of these
two parameters with fixed thresholds in some Cartesian graph products including
cycles and complete graphs.

Combinatorics