Nearly perfect sets in products of graphs

The study of nearly perfect sets in graphs was initiated in [2]. Let $S \subseteq V(G)$. We say that $S$ is a nearly perfect set (or is nearly perfect) in $G$ if every vertex in $V(G) - S$ is adjacent to at most one vertex in $S$. A nearly perfect set $S$ in $G$ is called maximal if for every vertex $u \in V(G) - S, S \cup \{u\}$ is not nearly perfect in $G$. The minimum cardinality of a maximal nearly perfect set is denoted by $n_{p}(G)$. It is our purpose in this paper to determine maximal nearly perfect sets in two well-known products of two graphs, i.e. in the Cartesian product and in the strong product. Lastly, we give upper bounds of $n_{p}(G1 \times G2)$ and $n_{p}(G1 \otimes G2)$, for some special graphs $G1,G2$.