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 $$S\subseteq V(G)$$ is said to be a 2-metric generator for G if and only if for any pair of different vertices $$u,v\in V(G)$$ , there exist at least two vertices $$w_1,w_2\in S$$ such that $$d_G(u,w_i)\ne d_G(v,w_i)$$ , for every $$i\in \{1,2\}$$ , 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 $$S\subseteq V(G)$$ is a 2-adjacency generator for G, if for every two vertices $$x,y\in V(G)$$ there exist at least two vertices $$w_1,w_2\in S$$ , such that $$d_{G,2}(x,w_i)\ne d_{G,2}(y,w_i)$$ for every $$i\in \{1,2\}$$ . 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 $$G\circ H$$ of two graphs G and H. Specifically, we show that $$\dim _2(G\circ H)=n\cdot {\mathrm {adim}}_2(H)+f(G,H),$$ where $$f(G,H)\ge 0$$ , and determine all the possible values of f(G, H).