k-metric resolvability in graphs

A set S of vertices of a graph G is said to be a k-metric generator for G if for any u,v∈V(G), u≠v, there exists Suv⊆S such that |Suv|≥k and for every w∈Suv, dG(u,w)≠dG(v,w). A metric generator of minimum cardinality is called a k-metric basis and its cardinality the k-metric dimension of G. We give a necessary and sufficient condition for the existence of a k-metric basis of a graph and we obtain several results on the k-metric dimension.