Revisiting the outer-weakly convex domination number in graph products

Abstract: Let $G = (V, E)$ be a simple undirected graph. A set $C \subseteq V(G)$ is weakly convex of graph $G$ if for every two vertices $u,v\in G$, there exists a $u-v$ geodesic whose vertices are in $C$. A set $C \subseteq V$ is an outer-weakly convex dominating set if it is dominating set and every vertex not in $C$ is adjacent to some vertex in $C$ and a set $V(G)\setminus C$ is weakly convex. The outer-weakly convex domination number of graph $G$, denoted by $\widetilde{ \gamma}_{wcon}(G)$, is the minimum cardinality of an outer-weakly convex dominating vertex set of graph $G$. In this paper, we determined the outer-weakly convex domination number of two graphs under the cartesian, strong and lexicographic products, and discuss some important combinatorial findings.