On graphs with the maximum edge metric dimension

An edge metric generator of a connected graph G is a vertex subset S for which every two distinct edges of G have distinct distance to some vertex of S, where the distance between a vertex v and an edge e is defined as the minimum of distances between v and the two endpoints of e in G. The smallest cardinality of an edge metric generator of G is the edge metric dimension, denoted by dime(G). It follows that 1≤dime(G)≤n−1 for any n-vertex graph G. A graph whose edge metric dimension achieves the upper bound is topful. In this paper, the structure of topful graphs is characterized, and many necessary and sufficient conditions for a graph to be topful are obtained. Using these results we design an O(n3) time algorithm which determines whether a graph of order n is topful or not. Moreover, we describe and address an interesting class of topful graphs whose super graphs obtained by adding one edge are not topful.