Simultaneous Resolvability in Families 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 $d_{G,2}(x,y)=\min\{d_{G}(x,y),2\}$.