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.

Let G = (V, E) be a connected graph. A vertex w ∈ V distinguishes two elements (vertices or edges) x, y ∈ E ∪ V if d G (w, x) = d G (w, y). A set S of vertices in a connected graph G is a mixed metric generator for G if every two elements (vertices or edges) of G are distinguished by some vertex of S. The smallest cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by mdim(G). In this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.

Let G = (V, E) be a connected graph. A vertex w ∈ V distinguishes two elements (vertices or edges). A set S of vertices in a connected graph G is a mixed metric generator for G if every two elements (vertices or edges) of G are distinguished by some vertex of S. The smallest cardinality of a mixed metric generator for G is called the mixed metric dimension and is denoted by mdim(G). In this paper we consider the structure of mixed metric generators and characterize graphs for which the mixed metric dimension equals the trivial lower and upper bounds. We also give results about the mixed metric dimension of some families of graphs and present an upper bound with respect to the girth of a graph. Finally, we prove that the problem of determining the mixed metric dimension of a graph is NP-hard in the general case.

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The metric (resp. edge metric or mixed metric) dimension of a graph G, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of G by using a vector of distances to this set. In this note we show two unexpected results on hypercube graphs. First, we show that the metric and edge metric dimension of Q d differ by only one for every integer d. In particular, if d is odd, then the metric and edge metric dimensions of Q d are equal. Second, we prove that the metric and mixed metric dimensions of the hypercube Q d are equal for every d ≥ 3. We conclude the paper by conjecturing that all these three types of metric dimensions of Q d are equal when d is large enough.

Let G = (V, E) be a graph. A set of vertices A is an incidence generator for G if for any two distinct edges e, f ∈ E(G) there exists a vertex from A which is an endpoint of either e or f . The smallest cardinality of an incidence generator for G is called the incidence dimension and is denoted by dim I (G). A set of vertices P is a 2-packing if the distance between any pair of distinct vertices from P is greater than two. The largest cardinality of a 2-packing of G is the packing number of G and is denoted by ρ(G). The incidence dimension of graphs is introduced and studied in this article, and we emphasize in the closed relationship between dim I (G) and ρ(G). We first note that the complement of any 2-packing in a graph G is always an incidence generator for G, and further show that either dim I (G) = ρ(G) or dim I (G) = ρ(G) − 1 for any graph G. In addition, we also prove that the problem of determining the incidence dimension of a graph is NP-complete, and present some bounds for it.

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Abstract. Let G = (V,E) be a graph. A set of vertices A is an incidence generator for G if for any two distinct edges e,f ∈ E(G) there exists a vertex from A which is an endpoint of either e or f. The smallest cardinality of an incidence generator for G is called the incidence dimension and is denoted by dimI(G). A set of vertices P ⊆ V (G) is a 2-packing of G if the distance in G between any pair of distinct vertices from P is larger than two. The largest cardinality of a 2-packing of G is the packing number of G and is denoted by ρ(G). In this article, the incidence dimension is introduced and studied. The given results show a close relationship between dimI(G) and ρ(G). We rst note that the complement of any 2-packing in graph G is an incidence generator for G, and further show that either dimI(G) = ρ(G) or dimI(G) = ρ(G)−1 for any graph G. In addition, we present some bounds for dimI(G) and prove that the problem of determining the incidence dimension of a graph is NP-hard.