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

A set of vertices \(W\) \emph{locally resolves} 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 \emph{local metric dimension} 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 \(n \geq 5\).