The Simultaneous Metric Dimension of Families Composed by Lexicographic Product Graphs

Let $\mathcal{G}$ be a graph family defined on a common (labeled) vertex set $V$. A set $S\subseteq V$ is said to be a simultaneous metric generator for $\mathcal{G}$ if for every $G\in \mathcal{G}$ and every pair of different vertices $u,v\in V$ there exists $s\in S$ such that $d_G(s,u)\mathrm{\ne }{d}_G(s,v)$, where $d_G$ denotes the geodesic distance. A simultaneous adjacency generator for $\mathcal{G}$ is a simultaneous metric generator under the metric $d_{G,2}(x,y)=\min \{d_G(x,y),2\}$. A minimum cardinality simultaneous metric (adjacency) generator for $\mathcal{G}$ is a simultaneous metric (adjacency) basis, and its cardinality the simultaneous metric (adjacency) dimension of $\mathcal{G}$. Based on the simultaneous adjacency dimension, we study the simultaneous metric dimension of families composed by lexicographic product graphs.