As a generalization of the concept of a metric basis, this article introduces
the notion of k-metric basis in graphs. Given a connected graph G=(V,E), a
set S⊆V is said to be a k-metric generator for G if the elements
of any pair of different vertices of G are distinguished by at least k
elements of S, i.e., for any two different vertices u,v∈V, there exist
at least k vertices w1,w2,...,wk∈S such that dG(u,wi)=dG(v,wi) for every i∈{1,...,k}. A metric generator of minimum
cardinality is called a k-metric basis and its cardinality the k-metric
dimension of G. A connected graph G is k-metric dimensional if k is the
largest integer such that there exists a k-metric basis for G. We give a
necessary and sufficient condition for a graph to be k-metric dimensional and
we obtain several results on the k-metric dimension