The Simultaneous Metric Dimension of Families Composed by Lexicographic Product Graphs

TL;DR

This paper investigates the minimum size of vertex sets that can distinguish all pairs of vertices across a family of lexicographic product graphs using metric and adjacency-based distances.

Contribution

It introduces the concept of simultaneous metric and adjacency dimensions for graph families and analyzes these parameters specifically for lexicographic product graph families.

Findings

Defined simultaneous metric and adjacency generators for graph families.

Established bounds and exact values for the simultaneous metric dimension of lexicographic product families.

Extended the understanding of metric-based graph invariants in complex graph constructions.

Abstract

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 ${\cal G}$ if for every $G\in {\cal G}$ 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 ${\cal G}$ 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 ${\cal G}$ is a simultaneous metric (adjacency) basis, and its cardinality the simultaneous metric (adjacency) dimension of ${\cal G}$. Based on the simultaneous adjacency dimension, we study the simultaneous metric dimension of families composed by lexicographic product graphs.