The k-metric dimension

For an undirected graph \(G=(V,E)\), a vertex \(\tau \in V\) separates vertices u and v (where \(u,v\in V\), \(u\ne v\)) if their distances to \(\tau \) are not equal. Given an integer parameter \(k \ge 1\), 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 \(\tau _{1},\tau _{2},\ldots ,\tau _{k}\in L\), 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 cost, and the goal is to find a landmark set with minimal total cost. We generalize the problem for \(k \ge 2\), 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 \(k \ge 2\). We present algorithms for these cases, thus demonstrating the difference between the two new models, and the differences between the cases \(k=1\) and \(k \ge 2\).