The kk-metric dimension of the lexicographic product of graphs

Given a simple and connected graph G=(V,E)G=(V,E), and a positive integer kk, a set S⊆VS⊆V is said to be a kk-metric generator for GG, if for any pair of different vertices u,v∈Vu,v∈V, there exist at least kk vertices w1,w2,…,wk∈Sw1,w2,…,wk∈S such that dG(u,wi)≠dG(v,wi)dG(u,wi)≠dG(v,wi), for every i∈{1,…,k}i∈{1,…,k}, where dG(x,y)dG(x,y) denotes the distance between xx and yy. The minimum cardinality of a kk-metric generator is the kk-metric dimension of GG. A set S⊆VS⊆V is a kk-adjacency generator for GG if any two different vertices x,y∈V(G)x,y∈V(G) satisfy |((NG(x)▿NG(y))∪{x,y})∩S|≥k, where NG(x)▿NG(y) is the symmetric difference of the neighborhoods of xx and yy. The minimum cardinality of any kk-adjacency generator is the kk-adjacency dimension of GG. In this article we obtain tight bounds and closed formulae for the kk-metric dimension of the lexicographic product of graphs in terms of the kk-adjacency dimension of the factor graphs.