Mixed metric dimension of graphs

Let G = (V, E) be a connected graph. A vertex w is an element of V distinguishes two elements (vertices or edges) x, y is an element of E boolean OR V if d(G)(w, x) not equal d(G)(w, y). A set S of vertices in a connected graph G is a mixed metric generator for G if every two distinct 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 dim(m) (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. (C) 2017 Elsevier Inc. All rights reserved.