On Fault-Tolerant Resolvability of Double Antiprism and its related Graphs

Abstract

For a connected graph $\Gamma =(V,E)$, a subset $R$ of ordered vertices in $V$ is said to be a resolving set in $\Gamma$, if the vector of distances to the vertices in $R$ is unique for each $u^i\in V(\Gamma )$. The metric dimension of $\Gamma$ is the minimum cardinality of such a set $R$. If $R\setminus \{u^i\}$ is still a resolving set $\forall$ $u^i\in R$, then $R$ is called a fault-tolerant resolving set (FTRS) for $\Gamma$ and its least cardinality is the fault-tolerant metric dimension (FTMD) of $\Gamma$. In this article, we introduce the concept of an independent fault-tolerant resolving set (IFTRS) and investigate it for several well-known graphs. We also show that the FTMD is four for three closely related families of convex polytopes available in the literature (viz., double antiprism ${\mathbb{A}}_n$, $S_n$, and $T_n$).

10 pages, 6 figures