  For an undirected graph G=(V,E), a vertex x \in V separates vertices u and v
(where u,v \in V, u \neq v) if their distances to x are not equal. Given an
integer parameter k \geq 1, a set of vertices L \subseteq V is a feasible
solution if for every pair of distinct vertices, u,v, there are at least k
distinct vertices x_1,x_2,...,x_k \in L each separating u and v. Such a
feasible solution is called a "landmark set", and the k-metric dimension of a
graph is the minimal cardinality of a landmark set for the parameter k. The
case k=1 is a classic problem, where in its weighted version, each vertex v has
a non-negative weight, and the goal is to find a landmark set with minimal
total weight. We generalize the problem for k \geq 2, introducing two models,
and we seek for solutions to both the weighted version and the unweighted
version of this more general problem. In the model of all-pairs (AP), k
separations are needed for every pair of distinct vertices of V, while in the
non-landmarks model (NL), such separations are required only for pairs of
distinct vertices in V \setminus L.
  We study the weighted and unweighted versions for both models (AP and NL),
for path graphs, complete graphs, complete bipartite graphs, and complete wheel
graphs, for all values of k \geq 2. We present algorithms for these cases, thus
demonstrating the difference between the two new models, and the differences
between the cases k=1 and k \geq 2.
