On Local Metric Dimension of $(n-3)$-regular Graph

Abstract

A set of vertices $W$ $locallyresolves$ a graph $G$ if every pair of adjacent vertices is uniquely determined by its coordinate of distances to the vertices in $W$. The minimum cardinality of a local resolving set of $G$ is called the $localmetricdimension$ of $G$. A graph $G$ is called a $k$-regular graph if every vertex of $G$ is adjacent to $k$ other vertices of $G$. In this paper, we determine the local metric dimension of an $(n-3)$-regular graph $G$ of order $n$, where $n\ge 5$.