The local edge metric dimension of graph

Abstract

In this paper, we introduce a new notion of graph theory study, namely a local edge metric dimension. It is a natural extension of metric dimension concept. dG (e,v) = min{d(x,v),d(y,v)} is the distance between the vertex v and the edge xy in graph G. A non empty set S V is an edge metric generator for G if for any two edges e1, e2 E there is a vertex k S such that dG(k,e1 ≠ dG(k,e2). The minimum cardinality of edge metric generator for G is called as edge metric dimension of G, denoted by dimE (G). The local edge metric dimension of G, denoted by dimE (G), is a local edge metric generator of G if r(xk|S)≠ r(yk|S) for every pair xk,ky of adjacent edges of G. Our concern in this paper is investigating some results of local edge metric dimension on some graphs.