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 $$d_{G}(s,u)\ne d_{G}(s,v)$$ , 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.