Simultaneous Resolvability in Families of Corona Product Graphs

Let $\mathcal{G}$ be a graph family defined on a common vertex set $V$ and let $d$ be a distance defined on every graph $G\in \mathcal{G}$. A set $S\subset V$ is said to be a simultaneous metric generator for $\mathcal{G}$ if for every $G\in {\cal G}$andeverypairofdifferentvertices$u,v\in V$thereexists$s\in S$suchthat$d(s,u)\ne d(s,v)$\mathrm{.}Thesimultaneousmetricdimensionof${\cal G}$isthesmallestinteger$k$ such that there is a simultaneous metric generator for $\mathcal{G}$ 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 $G\in \mathcal{G}$, namely, the geodesic distance $d_G$ and the distance $d_{G,2}:V\times V\rightarrow $\mathbb{N}$\cup {0}$definedas$d_{G,2}(x,y)=\min{d_{G}(x,y),2}$.