Fault Tolerance for Metric Dimension and Its Variants

Publication Date

1-16-2026

Document Type

Article

Publication Title

Discrete Applied Mathematics

Volume

383

DOI

10.1016/j.dam.2026.01.009

First Page

339

Last Page

354

Abstract

Hernando et al. (2008) introduced the fault-tolerant metric dimension ftdim(G), which is the size of the smallest resolving set S of a graph G such that S−s is also a resolving set of G for every s∈S. They found an upper bound ftdim(G)≤dim(G)(1+2⋅5dim(G)−1), where dim(G) denotes the standard metric dimension of G. It was unknown whether there exists a family of graphs where ftdim(G) grows exponentially in terms of dim(G), until recently when Knor et al. (2024) found a family with ftdim(G)=dim(G)+2dim(G)−1 for any possible value of dim(G). We improve the upper bound on fault-tolerant metric dimension by showing that ftdim(G)≤dim(G)(1+3dim(G)−1) for every connected graph G. Moreover, we find an infinite family of connected graphs Jk such that dim(Jk)=k and ftdim(Jk)≥3k−1−k−1 for each positive integer k. Together, our results show that limk→∞maxG:dim(G)=klog3(ftdim(G))k=1. In addition, we consider the fault-tolerant edge metric dimension ftedim(G) and bound it with respect to the edge metric dimension edim(G), showing that limk→∞maxG:edim(G)=klog2(ftedim(G))k=1. We also obtain sharp extremal bounds on fault-tolerance for adjacency dimension and k-truncated metric dimension. Furthermore, we obtain sharp bounds for some other extremal problems about metric dimension and its variants. In particular, we prove an equivalence between an extremal problem about edge metric dimension and an open problem of Erdős and Kleitman (1974) in extremal set theory.

Funding Number

113-2115-M-008-006-MY3

Funding Sponsor

Ministry of Science and Technology

Keywords

Edge metric dimension, Fault tolerance, Local metric dimension, Maximum degree, Metric dimension, Truncated metric dimension

Department

Mathematics and Statistics

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