On the edge metric dimension of graphs

Abstract

Let G = (V, E) be a connected graph of order n. S ⊆ V is an edge metric generator of G if any pair of edges in E can be distinguished by some element of S . The edge metric dimension edim(G) of a graph G is the least size of an edge metric generator of G. In this paper, we give the characterization of all connected bipartite graphs with edim = n − 2, which partially answers an open problem of Zubrilina (2018). Furthermore, we also give a sufficient and necessary condition for edim(G) = n − 2, where G is a graph with maximum degree n − 1. In addition, the relationship between the edge metric dimension and the clique number of a graph G is investigated by construction.