On the weak k-metric dimension of Hamming graphs

Given a connected graph G, a set of vertices X⊂ V(G) is a weak k-resolving set of G if for each two vertices y,z∈ V(G), the sum of the values |dG(y,x)-dG(z,x)| over all x∈ X is at least k, where dG(u,v) stands for the length of a shortest path between u and v. The cardinality of a smallest weak k-resolving set of G is the weak k-metric dimension of G, and is denoted by wdimₖ(G). In this paper, wdimₖ(Kₙ\,\,Kₙ) is determined for every n≥ 3 and every 2≤ k≤ 2n. An improvement of a known integer linear programming formulation for this problem is developed and implemented for the graphs Kₙ\,\,Kₘ. Conjectures regarding these general situations are posed.