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12 pages, 918 KiB

Open AccessArticle

Fault-Tolerant Edge Metric Dimension of Zero-Divisor Graphs of Commutative Rings

by Omaima Alshanquiti, Malkesh Singh and Vijay Kumar Bhat

Axioms 2025, 14(7), 499; https://doi.org/10.3390/axioms14070499 - 26 Jun 2025

Viewed by 255

In recent years, the intersection of algebraic structures and graph-theoretic concepts has developed significant interest, particularly through the study of zero-divisor graphs derived from commutative rings. Let Z*(S) be the set of non-zero zero divisors of a finite commutative ring [...] Read more.

In recent years, the intersection of algebraic structures and graph-theoretic concepts has developed significant interest, particularly through the study of zero-divisor graphs derived from commutative rings. Let Z*(S) be the set of non-zero zero divisors of a finite commutative ring S with unity. Consider a graph Γ(S) with vertex set V(Γ) = Z*(S), and two vertices in Γ(S) are adjacent if and only if their product is zero. This graph Γ(S) is known as zero-divisor graph of S. Zero-divisor graphs provide a powerful bridge between abstract algebra and graph theory. The zero-divisor graphs for finite commutative rings and their minimum fault-tolerant edge-resolving sets are studied in this article. Through analytical and constructive techniques, we highlight how the algebraic properties of the ring influence the edge metric structure of its associated graph. In addition to this, the existence of a connected graph G having a resolving set of cardinality of 2n + 2 from a star graph

K_{1,2 n}

, is studied. Full article

(This article belongs to the Special Issue Recent Developments in Graph Theory)

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14 pages, 280 KiB

Open AccessArticle

Fault-Tolerant Metric Dimension in Carbon Networks

by Kamran Azhar, Asim Nadeem and Yilun Shang

Foundations 2025, 5(2), 13; https://doi.org/10.3390/foundations5020013 - 16 Apr 2025

Viewed by 756

Abstract

In this paper, we study the fault-tolerant metric dimension in graph theory, an important measure against failures in unique vertex identification. The metric dimension of a graph is the smallest number of vertices required to uniquely identify every other vertex based on their [...] Read more.

In this paper, we study the fault-tolerant metric dimension in graph theory, an important measure against failures in unique vertex identification. The metric dimension of a graph is the smallest number of vertices required to uniquely identify every other vertex based on their distances from these chosen vertices. Building on existing work, we explore fault tolerance by considering the minimal number of vertices needed to ensure that all other vertices remain uniquely identifiable even if a specified number of these vertices fails. We compute the fault-tolerant metric dimension of various chemical graphs, namely fullerenes, benzene, and polyphenyl graphs. Full article

(This article belongs to the Section Mathematical Sciences)

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18 pages, 367 KiB

Open AccessArticle

Optimal Fault-Tolerant Resolving Set of Power Paths

by Laxman Saha, Rupen Lama, Bapan Das, Avishek Adhikari and Kinkar Chandra Das

Mathematics 2023, 11(13), 2868; https://doi.org/10.3390/math11132868 - 26 Jun 2023

Viewed by 1134

Abstract

In a simple connected undirected graph G, an ordered set R of vertices is called a resolving set if for every pair of distinct vertices u and v, there is a vertex

w \in R

such that [...] Read more.

In a simple connected undirected graph G, an ordered set R of vertices is called a resolving set if for every pair of distinct vertices u and v, there is a vertex

w \in R

such that

d (u, w) \neq d (v, w)

. A resolving set F for the graph G is a fault-tolerant resolving set if for each

v \in F

,

F ∖ {v}

is also a resolving set for G. In this article, we determine an optimal fault-resolving set of r-th power of any path

P_{n}

when

n \geq r (r - 1) + 2

. For the other values of n, we give bounds for the size of an optimal fault-resolving set. We have also presented an algorithm to construct a fault-tolerant resolving set of

P_{m}^{r}

from a fault-tolerant resolving set of

P_{n}^{r}

where

m < n

. Full article

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16 pages, 360 KiB

Open AccessArticle

Optimal Multi-Level Fault-Tolerant Resolving Sets of Circulant Graph C(n : 1, 2)

by Laxman Saha, Bapan Das, Kalishankar Tiwary, Kinkar Chandra Das and Yilun Shang

Mathematics 2023, 11(8), 1896; https://doi.org/10.3390/math11081896 - 17 Apr 2023

Cited by 6 | Viewed by 1423

Abstract

Let

G = (V (G), E (G))

be a simple connected unweighted graph. A set

R \subset V (G)

is called a fault-tolerant resolving set with the tolerance level k if the cardinality of [...] Read more.

Let

G = (V (G), E (G))

be a simple connected unweighted graph. A set

R \subset V (G)

is called a fault-tolerant resolving set with the tolerance level k if the cardinality of the set

S_{x, y} = {w \in R : d (w, x) \neq d (w, y)}

is at least k for every pair of distinct vertices

x, y

of G. A k-level metric dimension refers to the minimum size of a fault-tolerant resolving set with the tolerance level k. In this article, we calculate and determine the k-level metric dimension for the circulant graph

C (n : 1, 2)

for all possible values of k and

n .

The optimal fault-tolerant resolving sets with k tolerance are also delineated. Full article

(This article belongs to the Special Issue Applications of Algebraic Graph Theory and Its Related Topics)

13 pages, 535 KiB

Open AccessArticle

The Application of Fault-Tolerant Partition Resolvability in Cycle-Related Graphs

by Kamran Azhar, Sohail Zafar, Agha Kashif, Amer Aljaedi and Umar Albalawi

Appl. Sci. 2022, 12(19), 9558; https://doi.org/10.3390/app12199558 - 23 Sep 2022

Cited by 5 | Viewed by 1815

Abstract

The concept of metric-related parameters permeates all of graph theory and plays an important role in diverse networks, such as social networks, computer networks, biological networks and neural networks. The graph parameters include an incredible tool for analyzing the abstract structures of networks. [...] Read more.

The concept of metric-related parameters permeates all of graph theory and plays an important role in diverse networks, such as social networks, computer networks, biological networks and neural networks. The graph parameters include an incredible tool for analyzing the abstract structures of networks. An important metric-related parameter is the partition dimension of a graph holding auspicious applications in telecommunication, robot navigation and geographical routing protocols. A fault-tolerant resolving partition is a preference for the concept of a partition dimension. A system is fault-tolerant if failure of any single unit in the originally used chain is replaced by another chain of units not containing the faulty unit. Due to the optimal fault tolerance, cycle-related graphs have applications in network analysis, periodic scheduling and surface reconstruction. In this paper, it is shown that the partition dimension (PD) and fault-tolerant partition dimension (FTPD) of cycle-related graphs, including kayak paddle and flower graphs, are constant and free from the order of these graphs. More explicitly, the FTPD of kayak paddle and flower graphs is four, whereas the PD of flower graphs is three. Finally, an application of these parameters in a scenario of installing water reservoirs in a locality has also been furnished in order to verify our findings. Full article

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13 pages, 461 KiB

Open AccessArticle

Fault Tolerant Addressing Scheme for Oxide Interconnection Networks

by Asim Nadeem, Agha Kashif, Sohail Zafar, Amer Aljaedi and Oluwatobi Akanbi

Symmetry 2022, 14(8), 1740; https://doi.org/10.3390/sym14081740 - 21 Aug 2022

Cited by 2 | Viewed by 2137

Abstract

The symmetry of an interconnection network plays a key role in defining the functioning of a system involving multiprocessors where thousands of processor-memory pairs known as processing nodes are connected. Addressing the processing nodes helps to create efficient routing and broadcasting algorithms for [...] Read more.

The symmetry of an interconnection network plays a key role in defining the functioning of a system involving multiprocessors where thousands of processor-memory pairs known as processing nodes are connected. Addressing the processing nodes helps to create efficient routing and broadcasting algorithms for the multiprocessor interconnection networks. Oxide interconnection networks are extracted from the silicate networks having applications in multiprocessor systems due to their symmetry, smaller diameter, connectivity and simplicity of structure, and a constant number of links per node with the increasing size of the network can avoid overloading of nodes. The fault tolerant partition basis assigns unique addresses to each processing node in terms of distances (hops) from the other subnets in the network which work in the presence of faults. In this manuscript, the partition and fault tolerant partition resolvability of oxide interconnection networks have been studied which include single oxide chain networks (

S O X C N

), rhombus oxide networks (

R H O X N

) and regular triangulene oxide networks (

R T O X N

). Further, an application of fault tolerant partition basis in case of region-based routing in the networks is included. Full article

(This article belongs to the Special Issue Graph Theory and Its Applications)

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16 pages, 785 KiB

Open AccessArticle

On Resolvability- and Domination-Related Parameters of Complete Multipartite Graphs

by Sakander Hayat, Asad Khan and Yubin Zhong

Mathematics 2022, 10(11), 1815; https://doi.org/10.3390/math10111815 - 25 May 2022

Cited by 14 | Viewed by 2438

Abstract

Graphs of order n with fault-tolerant metric dimension n have recently been characterized.This paper points out an error in the proof of this characterization. We show that the complete multipartite graphs also have the fault-tolerant metric dimension n, which provides an infinite [...] Read more.

Graphs of order n with fault-tolerant metric dimension n have recently been characterized.This paper points out an error in the proof of this characterization. We show that the complete multipartite graphs also have the fault-tolerant metric dimension n, which provides an infinite family of counterexamples to the characterization. Furthermore, we find exact values of the metric, edge metric, mixed-metric dimensions, the domination number, locating-dominating number, and metric-locating-dominating number for the complete multipartite graphs. These results generalize various results in the literature from complete bipartite to complete multipartite graphs. Full article

(This article belongs to the Special Issue Graph Theory and Applications)

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16 pages, 344 KiB

Open AccessArticle

Fault-Tolerant Metric Dimension of Circulant Graphs

by Laxman Saha, Rupen Lama, Kalishankar Tiwary, Kinkar Chandra Das and Yilun Shang

Mathematics 2022, 10(1), 124; https://doi.org/10.3390/math10010124 - 1 Jan 2022

Cited by 20 | Viewed by 2485

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 [...] Read more.

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}, \dots, s_{k}} \subset V (G)

is called a resolving set for G if, for any two distinct vertices

u, v \in V (G)

, there is a vertex

s_{i} \in S

such that

d (u, s_{i}) \neq 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 \geq 22

. Full article

14 pages, 306 KiB

Open AccessArticle

Edge Metric and Fault-Tolerant Edge Metric Dimension of Hollow Coronoid

by Ali N. A. Koam, Ali Ahmad, Muhammad Ibrahim and Muhammad Azeem

Mathematics 2021, 9(12), 1405; https://doi.org/10.3390/math9121405 - 17 Jun 2021

Cited by 24 | Viewed by 2585

Abstract

Geometric arrangements of hexagons into six sides of benzenoids are known as coronoid systems. They are organic chemical structures by definition. Hollow coronoids are divided into two types: primitive and catacondensed coronoids. Polycyclic conjugated hydrocarbon is another name for them. Chemical mathematics piques [...] Read more.

Geometric arrangements of hexagons into six sides of benzenoids are known as coronoid systems. They are organic chemical structures by definition. Hollow coronoids are divided into two types: primitive and catacondensed coronoids. Polycyclic conjugated hydrocarbon is another name for them. Chemical mathematics piques the curiosity of scientists from a variety of disciplines. Graph theory has always played an important role in making chemical structures intelligible and useful. After converting a chemical structure into a graph, many theoretical and investigative studies on structures can be carried out. Among the different parameters of graph theory, the dimension of edge metric is the most recent, unique, and important parameter. Few proposed vertices are picked in this notion, such as all graph edges have unique locations or identifications. Different (edge) metric-based concept for the structure of hollow coronoid were discussed in this study. Full article

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19 pages, 376 KiB

Open AccessArticle

Fault-Tolerant Resolvability and Extremal Structures of Graphs

by Hassan Raza, Sakander Hayat, Muhammad Imran and Xiang-Feng Pan

Mathematics 2019, 7(1), 78; https://doi.org/10.3390/math7010078 - 14 Jan 2019

Cited by 60 | Viewed by 5356

Abstract

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n,

n - 1

, and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, [...] Read more.

In this paper, we consider fault-tolerant resolving sets in graphs. We characterize n-vertex graphs with fault-tolerant metric dimension n,

n - 1

, and 2, which are the lower and upper extremal cases. Furthermore, in the first part of the paper, a method is presented to locate fault-tolerant resolving sets by using classical resolving sets in graphs. The second part of the paper applies the proposed method to three infinite families of regular graphs and locates certain fault-tolerant resolving sets. By accumulating the obtained results with some known results in the literature, we present certain lower and upper bounds on the fault-tolerant metric dimension of these families of graphs. As a byproduct, it is shown that these families of graphs preserve a constant fault-tolerant resolvability structure. Full article

(This article belongs to the Special Issue Discrete Optimization: Theory, Algorithms, and Applications)

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