On the Local Metric Dimension of Line Graphs

Let G be a graph. For any uv is an element of E(G), if there exists w is an element of V(G) such that d(G)(w, u) =6 d(G)(w, v), we say that w resolving u, v. A set W of vertices in G is a local resolving set of G if there exists w is an element of W such that d(G)(w, v) =6 d(G)(w, u) for any uv is an element of E(G). The local metric dimension dim(e)(G) of G is the minimum cardinality of all the local resolving sets of G. In this paper, we study the relation between dim(e)(G) and dim(e)(L(G)). Furthermore, we construct a graph G(s,t) such that dim(e)(G(s,t)) = s and dim(e)(L(G(s,t))) = t. Finally, we investigate the local metric dimension of several special line graphs.