Differentiating Odd Dominating Sets in Graphs

Abstract

Let $G=(V(G),E(G))$ be a simple and undirected graph. A dominating set \linebreak $S\subseteq V(G)$ is called a \textit{differentiating odd dominating set} if for every vertex $v\in V(G)$, \linebreak $|N[v]\cap S|\equiv 1(mod\ 2)$ and $N_G[u]\cap S\neq N_G[v]\cap S$ for every two distinct vertices $u$ and $v$ of $V(G)$. The minimum cardinality of a differentiating odd dominating set of $G$, denoted by $\gamma_D^o(G)$, is called the \textit{differentiating odd domination number}. In this paper, we discuss \linebreak differentiating odd dominating set in some graphs and give relationships between the \linebreak 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.