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

Abstract

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).