Models for the k-metric dimension

For an undirected graph G = (V,E), a vertex τ ∈ V separates vertices u and v (where u, v ∈ V , u 6= v) if their distances to τ are not equal. Given an integer parameter k ≥ 1, a set of vertices L ⊆ V is a feasible solution if for every pair of distinct vertices, u, v, there are at least k distinct vertices τ1, τ2, . . . , τk ∈ 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 ≥ 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 \ 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 ≥ 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 ≥ 2.