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Non-isolated Resolving Number of Graphs Resulting from Corona Product

Last modified: 2017-07-05

#### Abstract

Let $G$ be a simple, finite and connected graph. For an ordered set $W=\{w_1,w_2,w_3,...,w_k\} \subset V(G)$ and a vertex $v \in V(G)$, the vector $r(v|W)=(d(v,w_1),d(v,w_2),\ldots,(v,w_k))$ is called the metric representation of $v$ with respect to $W$. If any two distinct vertices in $V(G)$ have distinct metric representations with respect to $W$, then $W$ is called a resolving set of $G$. The minimum cardinality of a resolving set of $G$ is called the metric dimension of $G$, denoted by dim$(G)$. A resolving set $W$ is called a non-isolated resolving set if the induced subgraph $[W]$ has no isolated vertices. The minimum cardinality of a non-isolated resolving set of $G$ is called the non-isolated resolving number of $G$, denoted by $nr(G)$. In this paper, we determine the non-isolated resolving number of graphs which are resulted from corona product.