Twin Vertices in Fault-Tolerant Metric Sets and Fault-Tolerant Metric Dimension of Multistage Interconnection Networks

TL;DR

This paper studies the fault-tolerant metric dimension of graphs, showing the importance of twin vertices, correcting previous characterizations, and analyzing specific network types like Butterfly and Benes networks.

Contribution

It proves that twin vertices are essential in fault-tolerant bases, corrects a previous characterization, and extends analysis to specific interconnection networks.

Findings

Twin vertices belong to every fault-tolerant basis.

Fault-tolerant metric dimension equals the number of vertices iff all vertices are twins.

Disproves previous conjectures related to fault-tolerant metric dimension.

Abstract

A set of vertices $S\subseteq V(G)$ is a basis or resolving set of a graph $G$ if for each $x,y\in V(G)$ there is a vertex $u\in S$ such that $d(x,u)\neq d(y,u)$. A basis $S$ is a fault-tolerant basis if $S\setminus \{x\}$ is a basis for every $x \in S$. The fault-tolerant metric dimension (FTMD) $\beta'(G)$ of $G$ is the minimum cardinality of a fault-tolerant basis. It is shown that each twin vertex of $G$ belongs to every fault-tolerant basis of $G$. As a consequence, $\beta'(G) = n(G)$ iff each vertex of $G$ is a twin vertex, which corrects a wrong characterization of graphs $G$ with $\beta'(G) = n(G)$ from [Mathematics 7(1) (2019) 78]. This FTMD problem is reinvestigated for Butterfly networks, Benes networks, and silicate networks. This extends partial results from [IEEE Access 8 (2020) 145435--145445], and at the same time, disproves related conjectures from the same paper.