Uniquely identifying the edges of a graph: The edge metric dimension

Abstract

Let G = (V, E) be a connected graph, let v ∈ V be a vertex and let e = uw ∈ E be an edge. The distance between the vertex v and the edge e is given by. A set S of vertices in a connected graph G is an edge metric generator for G if every two edges of G are distinguished by some vertex of S. The smallest cardinality of an edge metric generator for G is called the edge metric dimension and is denoted by edim(G). In this article we introduce the concept of edge metric dimension and initiate the study of its mathematical properties. We make a comparison between the edge metric dimension and the standard metric dimension of graphs while presenting some realization results concerning the edge metric dimension and the standard metric dimension of graphs. We prove that computing the edge metric dimension of connected graphs is NP-hard and give some approximation results. Moreover, we present some bounds and closed formulae for the edge metric dimension of several classes of graphs.

Mixed metric dimension of graphs

“…every vertex of the graph is identified by S or vice versa. However, as it proved in [16], this is further away from the reality, although there are several graph families in which such facts occur. In [16], among other results, some comparison between these two parameters above were discussed.…”

“…However, as it proved in [16], this is further away from the reality, although there are several graph families in which such facts occur. In [16], among other results, some comparison between these two parameters above were discussed. As a consequence of the study, families of graphs G, for which edim(G) < dim(G) or edim(G) = dim(G) or dim(G) < edim(G) hold were described.…”

“…Proof. From [3] and [16] we know that dim(K r,t ) = edim(K r,t ) = r + t − 2. So, by using (1) we have mdim(K r,t ) ≥ r + t − 2.…”

“…Let S be the set of all leaves of T and let x, y be any two distinct elements of T . From [17] and [16] is known that there are a metric basis and an edge metric basis which are both subsets of leaves in T . Thus, if x, y are either two vertices or two edges, then they are distinguished by S, which is formed by all leaves of T .…”

“…In connection with describing other new variants of metric generators in graph, very recently a parameter used to uniquely recognize the edges of the graph has been introduced in [16]. Roughly speaking, there was used a graph metric to identify each pair of edges by mean of distances to a fixed set of vertices.…”