Fault-tolerant resolvability of double antiprism and its related graphs

Abstract

Consider a graph, $\Gamma=(V,E)$, where $V(\Gamma)$ and $E(\Gamma)$ are referred to be its vertex and edge sets respectively. Two vertices $k_{1}$ and $k_{2}$ in $\Gamma$ are said to be resolved by a vertex $k$, if $d(k, k_1)\neq d(k, k_2)$ in $\Gamma$. Then, a subset $R\subseteq V(\Gamma)$ with this property, i.e., every pair of different vertices in $\Gamma$ can be resolved by at least one member of $R$, is said to be a resolving set (RS) for $\Gamma$. The smallest cardinality set $R$ with resolving characteristic is called the metric basis (MB) for $\Gamma$, and the MB set cardinality is the metric dimension (MD) for $\Gamma$, denoted by $dim_{v}(\Gamma)$. A resolving set $R_f$ for $\Gamma$ is said to have the property of fault-tolerance or said to be FTRS (fault-tolerant resolving set) if the property of resolving holds for $R_f-\{k\}$ for every $k$ in $R_f$. The FTMD of $\Gamma$ is the minimum cardinality of a FTRS, denoted $dim_{f}(\Gamma)$. These are also known as the resolvability parameters for the $\Gamma$. We introduce the concept of independence in FTRSs for graphs and derive several results and observations for the same in this manuscript. We also consider three almost similar families of infinite convex polytopes and investigate their FTRSs as well as FTMD.