Nonlocal metric dimension of some hierarchical product graphs

Let G and H be simple graphs and let S⊆V(G). A set S is called a nonlocal resolving set of G if each pair of non-adjacent vertices of G is distinguished by at least one vertex of S. The nonlocal metric dimension of G, denoted by dimnl⁡(G), is the minimum cardinality of such a set. This invariant is complementary to the classical metric dimension and to the local metric dimension: instead of distinguishing all vertex pairs or only adjacent vertex pairs, it isolates the information needed to distinguish non-adjacent pairs. This distinction is especially relevant for graph products in which the construction itself creates new non-adjacencies. In this paper, we study the nonlocal metric dimension of hierarchical product graphs. We first give a general lower bound for dimnl⁡(G(U)⊓H) in terms of dimnl⁡(H), and then discuss exact values for several families, including Kn(U)⊓H, K1,m(U)⊓Pn, and Pn(U)⊓K1,m.