Relationships Between the 2-Metric Dimension and the 2-Adjacency Dimension in the Lexicographic Product of Graphs

Given a connected simple graph (G=(V(G),E(G))), a set (Ssubseteq V(G)) is said to be a 2-metric generator for G if and only if for any pair of different vertices (u,vin V(G)), there exist at least two vertices (w_1,w_2in S) such that (d_G(u,w_i)ne d_G(v,w_i)), for every (iin {1,2}), where (d_G(x,y)) is the length of a shortest path between x and y. The minimum cardinality of a 2-metric generator is the 2-metric dimension of G, denoted by (dim _2(G)). The metric (d_{G,2}: V(G)times V(G)longmapsto {mathbb {N}}cup {0}) is defined as (d_{G,2}(x,y)=min {d_G(x,y),2}). Now, a set (Ssubseteq V(G)) is a 2-adjacency generator for G, if for every two vertices (x,yin V(G)) there exist at least two vertices (w_1,w_2in S), such that (d_{G,2}(x,w_i)ne d_{G,2}(y,w_i)) for every (iin {1,2}). The minimum cardinality of a 2-adjacency generator is the 2-adjacency dimension of G, denoted by ({mathrm {adim}}_2(G)). In this article, we obtain closed formulae for the 2-metric dimension of the lexicographic product (Gcirc H) of two graphs G and H. Specifically, we show that (dim _2(Gcirc H)=ncdot {mathrm {adim}}_2(H)+f(G,H),) where (f(G,H)ge 0), and determine all the possible values of f(G, H).