Simultaneous Resolvability in Families of Corona Product Graphs

Abstract

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 $G\in \mathcal{G}$ 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 $\mathcal{G}$ is the smallest integer $k$ 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 $d_{G,2}:V\times V\to \mathbb{N}\cup \{0\}$ defined as $d_{G,2}(x,y)=\min \{d_G(x,y),2\}$.

arXiv admin note: substantial text overlap with arXiv:1504.00492

References in corpus (2)

• The Simultaneous Metric Dimension of Graph Families
• The Simultaneous Metric Dimension of Families Composed by Lexicographic Product Graphs