On vertex resolvability of a circular ladder of nonagons

Abstract

Let $H=H(V,E)$ be a non-trivial simple connected graph with edge and vertex set $E(H)$ and $V(H)$, respectively. A subset $\mathbb{D}\subset V(H)$ with distinct vertices is said to be a vertex resolving set in $H$ if for each pair of distinct vertices $p$ and $q$ in $H$ we have $d(p,u)\neq d(q,u)$ for some vertex $u\in H$. A resolving set $H$ with minimum possible vertices is said to be a metric basis for $H$. The cardinality of metric basis is called the metric dimension of $H$, denoted by $\dim_{v}(H)$. In this paper, we prove that the metric dimension is constant and equal to $3$ for certain closely related families of convex polytopes.