Simultaneous Resolvability in Families of Corona Product Graphs

TL;DR

This paper investigates the minimal size of a vertex set that can distinguish all pairs of vertices across a family of corona product graphs using two specific distance measures, expanding understanding of simultaneous metric dimensions.

Contribution

It introduces the concept of simultaneous metric dimension for corona product graph families with two distance metrics, providing new bounds and characterizations.

Findings

Determined bounds for the simultaneous metric dimension in corona product graph families.

Characterized the impact of two distance measures on resolvability.

Provided exact values for specific classes of corona product graphs.

Abstract

Let ${\cal G}$ be a graph family defined on a common vertex set $V$ and let $d$ be a distance defined on every graph $G\in {\cal G}$. A set $S\subset V$ 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 $d(s,u)\ne d(s,v)$. The simultaneous metric dimension of ${\cal G}$ is the smallest integer $k$ such that there is a simultaneous metric generator for ${\cal G}$ of cardinality $k$. We study the simultaneous metric dimension of families composed by corona product graphs. Specifically, we focus on the case of two particular distances defined on every $G\in {\cal G}$, namely, the geodesic distance $d_G$ and the distance $d_{G,2}:V\times V\rightarrow \mathbb{N}\cup \{0\}$ defined as $d_{G,2}(x,y)=\min\{d_{G}(x,y),2\}$.