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Let $G$ be a simple graph with vertex set $V$ and edge set $E$. $ B_{G, NINC,\overline{ K_{q}}}$ $(G)$, known as boolean graph of $G$-first kind,simply denoted by $BG_{1}(G)$ is defined as the graph with vertex set $V\cup E$ and two vertices are adjacent if and only if they correspond to adjacent vertices in $G$ or to a vertex and an edge in $G$ such that the edge is not incident with the vertex. In this paper we give a bound for metric dimension of $BG_{1}(G)$ and also find expression for metric dimension of boolean graphs of Complete graphs and Star graphs. Finally, an algorithm for finding the metric dimension of $BG_{1}(G)$ is established.

Keywords and Phrases

Boolean graph $BG_{1}(G)$, metric dimension, resoliving set.

A.M.S. subject classification

05C48, 05C85.


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