Some Bounds on Co-isolated Locating Domination Number

Abstract

Let $G (V, E)$ be a simple, finite and undirected connected graph. A non-empty set $S\subseteq V$ of a graph G is a dominating set, if every vertex in $V-S$ is adjacent to atleast one vertex in S. A dominating set $S\subseteq V$ is called a locating dominating set, if for any two vertices $v, w\in V-S$, $N(v)\cap S\neq N(w)\cap S$. A locating dominating set $S\subseteq V$ is called a co-isolated locating dominating set (cild - set), if there exists atleast one isolated vertex in $\langle V-S\rangle$. The co-isolated locating domination number $\gamma_{cild}$ is the minimum cardinality of a co-isolated locating dominating set. In this paper, some bounds on co-isolated locating domination number are obtained. Also minimal cild - sets are characterized. Further the graphs for which $\gamma_{cild}$ to be $p-2$ are obtained.