Citation: Saha, L.; Lama, R.; Tiwary,
K.; Das, K.C.; Shang, Y. Fault-Tolerant
Metric Dimension of Circulant
Graphs . Mathematics 2022, 10, 124.
https://doi.org/10.3390/
math10010124
Academic Editor: Elena Guardo
Received: 29 November 2021
Accepted: 28 December 2021
Published: 1 January 2022
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mathematics
Article
Fault-Tolerant Metric Dimension of Circulant Graphs
Laxman Saha
1
, Rupen Lama
1
, Kalishankar Tiwary
2
, Kinkar Chandra Das
3,
* and Yilun Shang
4,
*
1
Department of Mathematics, Balurghat College, Balurghat 733101, India;
laxman.math@balurghatcollege.ac.in (L.S.); rupen.maths@balurghatcollege.ac.in (R.L.)
2
Department of Mathematics, Raiganj University, Raiganj 733134, India; kstiwary@raiganjuniversity.ac.in
3
Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
4
Department of Computer and Information Sciences, Northumbria University, Newcastle NE1 8ST, UK
* Correspondence: kinkardas2003@gmail.com or kinkar@skku.edu (K.C.D.);
yilun.shang@northumbria.ac.uk (Y.S.)
Abstract: Let G be a connected graph with vertex set V(G) and d(u, v) be the distance between the
vertices u and v. A set of vertices S = {s
1
, s
2
, ... , s
k
}⊂ V(G) is called a resolving set for G if, for any
two distinct vertices u, v ∈ V(G), there is a vertex s
i
∈ S such that d(u, s
i
) = d(v, s
i
). A resolving set
S for G is fault-tolerant if S \{x} is also a resolving set, for each x in S, and the fault-tolerant metric
dimension of G, denoted by β
′
(G), is the minimum cardinality of such a set. The paper of Basak
et al. on fault-tolerant metric dimension of circulant graphs C
n
(1, 2, 3) has determined the exact value
of β
′
(C
n
(1, 2, 3)). In this article, we extend the results of Basak et al. to the graph C
n
(1, 2, 3, 4) and
obtain the exact value of β
′
(C
n
(1, 2, 3, 4)) for all n ≥ 22.
Keywords: circulant graphs; resolving set; fault-tolerant resolving set; fault-tolerant metric dimension
1. Introduction
The distance between two vertices u and v, denoted by d
G
(u, v), is the length of the
shortest u − v path in a simple, undirected, connected graph G with the vertex set V(G)
and the edge set E(G). Whenever there is no possibility of confusion, we will simply write
d(u, v) instead of d
G
(u, v). A vertex z resolves two vertices x and y if d(x, z) = d(y, z). Let
S ⊂ V(G) be a set with m elements. The code of a vertex w with respect to S, denoted
by c(w|S), is the m-tuple c(w|S)=(d(w, s):s ∈ S). A set S is a resolving set if distinct
vertices have distinct codes, i.e., if c( x|S)= c(y|S) for all distinct x, y ∈ V(G). Equiva-
lently, S is said to be a resolving set for G if for every pair of distinct vertices x and y,
there is a s ∈ S such that c( x|S) = c(y|S). The metric dimension of G is the number
min
S
{|S|:S is a resolving set of G} and it is denoted by β(G).
Slater [1] and Harry et al. [2] have introduced the metric dimension of graphs. A metric
basis is a resolving set with the cardinality β(G). Some times metric bases elements may be
considered as censors, see [3]. We will not have enough knowledge to deal with the attacker
(fire, thief etc.) if one of the censors malfunctions. In order to overcome this kind of problems,
Hernando et al. have proposed the concept of fault-tolerant metric dimension in [4].
A resolving set S of a graph G is fault-tolerant if for each u ∈ S, S \{u} is also
a resolving set for G. The fault-tolerant metric dimension of G, denoted by β
′
(G), is the
minimum cardinality of a fault-tolerant resolving set. A fault-tolerant metric basis is a
fault-tolerant resolving set of order β
′
(G).
Determining a graph’s fault-tolerant metric dimension is a challenging combinatorial
problem with potential applications in sensor networks. It has only been tested for a
few simple graph families thus far. Hernendo et al. characterized the fault tolerant
resolving sets in a tree T in their introductory paper [4]. They have also furnished an
upper bound for the fault-tolerant metric dimension of an arbitrary graph G as β
′
(G)
β(G)(1 + 2 × 5
β(G)−1
). Saha [5] determined the fault-tolerant metric dimension of cube of
paths, and Javaid et al. [6] obtained β
′
(C
n
), where C
n
is a cycle of order n.
Mathematics 2022, 10, 124. https://doi.org/10.3390/math10010124 https://www.mdpi.com/journal/mathematics