$F_{3}$ -domination problem of graphs

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Given a graph $G$ and a set $S\subseteq V(G),$ a vertex $v$ is said to be $F_{3}$ -dominated by a vertex $w$ in $S$ if either $v=w,$ or $v\notin S$ and there exists a vertex $u$ in $V(G)-S$ such that $P:wuv$ is a path in $G$ . A set $S\subseteq V(G)$ is an $F_{3}$ -dominating set of $G$ if every vertex $v$ is $F_{3}$ -dominated by a vertex $w$ in $S.$ The $F_{3}$ -domination number of $G$ , denoted by $\gamma _{F_{3}}(G)$ , is the minimum cardinality of an $F_{3}$ -dominating set of $G$ . In this paper, we study the $F_{3}$ -domination of Cartesian product of graphs, and give formulas to compute the $F_{3}$ -domination number of $P_{m}\times P_{n}$ and $P_{m}\times C_{n}$ for special $m,n.$