On Locating-Dominating Set of Regular Graphs

Abstract

Let $G$ be a simple, connected, and finite graph. For every vertex $v\in V\left(G\right)$ , we denote by $N_G\left(v\right)$ the set of neighbours of $v$ in $G$ . The locating-dominating number of a graph $G$ is defined as the minimum cardinality of $W\subseteq V\left(G\right)$ such that every two distinct vertices $u,v\in V\left(G\right)\backslash W$ satisfies $\varnothing \ne N_G\left(u\right)\cap W\ne N_G\left(v\right)\cap W\ne \varnothing$ . A graph $G$ is called $k$ -regular graph if every vertex of $G$ is adjacent to $k$ other vertices of $G$ . In this paper, we determine the locating-dominating number of $k$ -regular graph of order $n$ , where $k=n-2$ or $k=n-3$ .