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Locating Dominating Set of $m$-shadowwing of Graphs

Last modified: 2017-07-05

#### Abstract

Let $G=(V,E)$ be a connected, undirected and simple graph. We define a set $D$ as a dominating set if for every vertex $u \in V - D$ is adjacent to some vertex $v \in D$. The domination number $\gamma(G)$ is the minimum cardinality of dominating set. A vertex set D in graph $G=(V,E)$ is called locating dominating set if for every pair of different vertex $u$ and $v$ in $V(G)-D$ which occupies $\emptyset \neq N(u) \cap D \neq N(v) \cap D$ where $N(u)$ is adjacency vertex set of u. The minimal cardinality of locating domination number is denoted by $\gamma_L (G)$. We present about the locating dominating set of $m$-shadowwing graph and some bounds of it. Let $G$ be simple and connected graph. Shadow graph is defined by graph which build by copying $G$ graph become $G'$ and $G"$. If $v'$ is vertex in $G'$ and $v"$ is vertex in $G"$, then every vertex $v'$ in $G'$ adjacent $v"$ in $G"$. $m$-shadowwing graph is shadow graph which denote by $D_m(G)$, has the number of copy $G$ as $m$.