Double total domination number of Cartesian product of paths

A vertex set $S$ of a graph $G$ is called a double total dominating set if every vertex in $G$ has at least two adjacent vertices in $S$. The double total domination number ${\gamma }_{\times 2,t}(G)$ of $G$ is the minimum cardinality over all the double total dominating sets in $G$. Let $G◻H$ denote the Cartesian product of graphs $G$ and $H$. In this paper, the double total domination number of Cartesian product of paths is discussed. We determine the values of ${\gamma }_{\times 2,t}(P_i◻P_n)$ for $i=2,3$, and give lower and upper bounds of ${\gamma }_{\times 2,t}(P_i◻P_n)$ for $i\ge 4$.