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

Given a connected simple graph $$G=(V(G),E(G))$$G=(V(G),E(G)), a set $$S\subseteq V(G)$$S⊆V(G) is said to be a 2-metric generator for G if and only if for any pair of different vertices $$u,v\in V(G)$$u,v∈V(G), there exist at least two vertices $$w_1,w_2\in S$$w1,w2∈S such that $$d_G(u,w_i)\ne d_G(v,w_i)$$dG(u,wi)≠dG(v,wi), for every $$i\in \{1,2\}$$i∈{1,2}, where $$d_G(x,y)$$dG(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 $$\dim _2(G)$$dim2(G). The metric $$d_{G,2}: V(G)\times V(G)\longmapsto {\mathbb {N}}\cup \{0\}$$dG,2:V(G)×V(G)⟼N∪{0} is defined as $$d_{G,2}(x,y)=\min \{d_G(x,y),2\}$$dG,2(x,y)=min{dG(x,y),2}. Now, a set $$S\subseteq V(G)$$S⊆V(G) is a 2-adjacency generator for G, if for every two vertices $$x,y\in V(G)$$x,y∈V(G) there exist at least two vertices $$w_1,w_2\in S$$w1,w2∈S, such that $$d_{G,2}(x,w_i)\ne d_{G,2}(y,w_i)$$dG,2(x,wi)≠dG,2(y,wi) for every $$i\in \{1,2\}$$i∈{1,2}. The minimum cardinality of a 2-adjacency generator is the 2-adjacency dimension of G, denoted by $${\mathrm {adim}}_2(G)$$adim2(G). In this article, we obtain closed formulae for the 2-metric dimension of the lexicographic product $$G\circ H$$G∘H of two graphs G and H. Specifically, we show that $$\dim _2(G\circ H)=n\cdot {\mathrm {adim}}_2(H)+f(G,H),$$dim2(G∘H)=n·adim2(H)+f(G,H), where $$f(G,H)\ge 0$$f(G,H)≥0, and determine all the possible values of f(G, H).