Consider an undirected and connected graph G = (V G , E G ), where V G and E G represent the set of vertices and the set of edges respectively. The concept of edge version of metric dimension and doubly resolving sets is based on the distances of edges in a graph. In this paper, we find the edge version of metric dimension and doubly resolving sets for the necklace graph.

The minimum edge version of metric basis is the smallest set [Formula: see text] of edges in a connected graph [Formula: see text] such that for every pair of edges [Formula: see text] [Formula: see text][Formula: see text] there exists an edge [Formula: see text] [Formula: see text][Formula: see text] for which [Formula: see text] [Formula: see text] [Formula: see text] holds. In this paper, the families of grid graphs and generalized prism graphs have been studied for edge version of metric dimension. Edge version of metric dimension is found to be constant for both families of graphs.

Graph theory is widely used to analyze the structure models in chemistry, biology, computer science, operations research and sociology. Molecular bonds, species movement between regions, development of computer algorithms, shortest spanning trees in weighted graphs, aircraft scheduling and exploration of diffusion mechanisms are some of these structure models. Let G = (V G , E G ) be a connected graph, where V G and E G represent the set of vertices and the set of edges respectively. The idea of the edge version of metric dimension is based on the distance of edges in a graph. Let R E G be the smallest set of edges in a connected graph G that forms a basis such that for every pair of edges e 1 , e 2 ∈ E G , there exists an edge e ∈ R E G for which d E G (e 1 , e) = d E G (e 2 , e) holds. In this paper, we show that the family of circulant graphs C n (1, 2) is the family of graphs with constant edge version of metric dimension.INDEX TERMS Line graph, Resolving sets, The edge version of metric dimension, Circulant graphs.