On monopoly and dynamic monopoly of Cartesian product of graphs with constant thresholds

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.