Differentiating Odd Dominating Sets in Graphs (European Journal of Pure and Applied Mathematics, 2024)
Let $G=(V(G),E(G))$ be a simple and undirected graph. A dominating set $S\subseteq V(G)$ is called a differentiating odd dominating set if for every vertex $v\in V(G)$, $|N[v]\cap S|\equiv 1(mod\ 2)$ and $N_G[u]\cap S eq N_G[v]\cap S$ for every two distinct vertices $u$ and $v$ of $V(G)$. The minimum cardinality of a differentiating odd dominating set of $G$, denoted by $γ_D^o(G)$, is called the differentiating odd domination number. In this paper, we discuss differentiating odd dominating set in some graphs and give relationships between the differentiating odd domination, odd domination, and differentiating-domination numbers. Moreover, we characterize the differentiating odd dominating sets in graphs resulting from join, corona, and lexicographic product of graphs and determine the differentiating odd domination numbers of these graphs.
Keywords: differenting, domination, odd dominating, differentiating-dominating, differentiating odd dominating