Strong and Weak Dominating Sets of Graphs Under Some Binary Operations (European Journal of Pure and Applied Mathematics, 2025)
A set $S$ of vertices of a graph $G$ is a strong (resp. weak) dominating set of $G$ if for every vertex $v$ of $G$ outside of $S$, there is a vertex $u$ inside of $S$ such that $u$ and $v$ are adjacent and $deg_G(v)\le deg_G(u)$ (resp. $deg_G(v)\ge deg_G(u)$). The minimum cardinality of a strong (resp. weak) dominating set is called the strong (resp. weak) domination number of $G$, and is denoted by $γ_s(G)$ (resp. $γ_w(G)$). In this paper, we characterize the strong and weak dominating sets of graphs under some binary operations. As a result, we also determine the exact values of or sharp bounds for the corresponding strong and weak domination numbers.
Keywords: strong dominating, weak dominating, complementary prism, join, corona, edge corona, lexicographic product