Revisiting the outer-weakly convex domination number in graph products

Abstract

Let $G = (V, E)$ be a simple undirected connected graph. A set $C \subseteq V(G)$ is weakly convex in $G$ 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 every vertex not in $C$ is adjacent to some vertex in $C$ and the set $V(G)\setminus C$ is weakly convex in $G$. 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 set of graph $G$. In this paper, we determine the outer-weakly convex domination number of two graphs under the Cartesian, strong and lexicographic products, and discuss some important combinatorial findings.