The local edge metric dimension of graph

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. d G ( 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 e 1 , e 2 ∈ E there is a vertex k ∈ S such that d G ( k , e 1 ≠ d G ( k , e 2 ) ) . The minimum cardinality of edge metric generator for G is called as edge metric dimension of G, denoted by dim E ( G ). The local edge metric dimension of G , denoted by dim E ( G ), is a local edge metric generator of G if r ( x k | S ) ≠ r ( y k | 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.