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

TL;DR

This paper introduces the concept of independent fault-tolerant resolving sets and determines the fault-tolerant metric dimension for certain convex polytopes, including double antiprism graphs.

Contribution

It defines the new concept of an independent fault-tolerant resolving set and calculates the fault-tolerant metric dimension for specific convex polytope graphs.

Findings

FTMD is four for double antiprism, S_n, and T_n graphs.

Introduces the concept of IFTRS for well-known graphs.

Provides new insights into fault-tolerant resolvability in graph theory.

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 $T_{n}$).