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On the local dominating set of \\ $Pn \trianglerighteq H$ graph

Last modified: 2017-07-05

#### Abstract

A subset $D$ of $V(G)$ is called a vertex dominating set of $G$ if every vertex not in $D$ is adjacent to some vertices in $D$. A graph $G=(V,E)$ is called a locating dominating set if for every two vertices $u,v \epsilon V(G)-D$, $ N(v)\cap D \neq \emptyset$. Locating dominating number $\gamma_L(G)$ is the minimum cardinality of a locating dominating set. The value of locating dominating number is $\gamma_L(G)\subseteq V(G)$. Edge comb product denoted by $G \trianglerighteq H$ is a graph obtained by taking one copy of $G$ and $|E(G)|$ copies of $H$ and grafting the $i$-th copy of $H$ at the edge $e$ to the $i$-th edge of $G$. This paper studies about locating dominating set in edge comb product graph $G \trianglerighteq H$ where $G$ is path graph $P_n$ and $H$ is any special graph.