Abstract
This study explores the fault-tolerant metric dimension (FTMD) of the para-line network, which is derived from the n-sunlet network, a significant category of networks created by subdividing and transforming cycle-based networks to simulate greater structural complexity. FTMD ensures robust vertex identification even when some nodes fail, a characteristic essential for fault-resilient systems like communication and sensor networks. By rigorously analyzing the structural attributes of these para-line networks, we demonstrate that the FTMD consistently behaves in a specific manner: it is equal to 3 when n=3 or 6, and 4 otherwise. Our approach includes constructive distance-vector analysis and combinatorial proofs to ensure the smallest possible size of the fault-tolerant resolving set (FTRS). These findings contribute both theoretically and practically, offering new perspectives on the development of resilient network topologies for practical applications such as IoT infrastructure, smart cities, and fault-tolerant distributed systems.
Keywords: Optimization, Artificial Intelligence (AI), Complex networks, Subdivision and para-line networks, n-sunlet network, Metric and fault-tolerant dimensions
Subject terms: Engineering, Mathematics and computing, Physics
Introduction
The metric dimension (MD) of a network is a fundamental parameter in graph theory, introduced to uniquely identify all vertices of a network based on their distances from a selected set of vertices. This concept, formalized by Slater1 and Harary and Melter2, has applications in diverse areas such as robotics, where it aids in navigation and localization tasks; network security, where MD supports the detection of intrusions and vulnerabilities, and bioinformatics, where it helps analyze genetic and molecular structures3.
An extension of this concept, the FTMD, ensures that the unique identification of vertices remains intact even when a fixed number of vertices or edges fail. FTMD has proven essential in the design of resilient systems such as communication networks4, where uninterrupted operations are required despite failures; in wireless sensor networks5, where sensor failures are inevitable for real-time monitoring and data collection; and in disaster management6, where maintaining critical routes during infrastructure breakdowns is vital.
Among various network families, the n-sunlet network, a network formed by attaching a pendant vertex to each vertex of a cycle, has attracted significant attention. Its hierarchical and hybrid structure is useful in modeling hub-and-spoke communication networks and distributed computing architectures. Extending this structure, the para-line network of an n-sunlet network incorporates additional edges, creating a richer topology for investigating fault-tolerant properties.
While recent advancements in FTMD have yielded insights into families such as Cartesian product networks7, circulant networks8, and hierarchical networks, the FTMD of para-line networks derived from n-sunlet networks remain unexplored. This study seeks to address this gap by analyzing the FTMD of para-line networks of n-sunlet networks, providing exact values, bounds, and structural properties. The findings are expected to contribute significantly to the theoretical understanding of FTMD and to practical applications in network optimization and system reliability.
Previous work
The concept of MD in graph theory was first introduced by Slater1 through the notion of leaves of trees, which laid the foundation for vertex resolvability in networks. Following this, Harary and Melter2 formally defined the MD of a network, establishing its significance in structural network analysis. Chartrand et al.9 further advanced the field by formalizing the concept of resolvability and studying its applications across different network classes. Poisson and Zhang10 extended these concepts to unicyclic networks, characterizing their MD and contributing to the broader study of resolvability in networks, while Chartrand and Zhang11 provided a comprehensive survey on resolvability, highlighting its theoretical and practical implications.
Furthermore, many network families have been explored concerning their MD. For instance, Javaid et al.12 focused on regular networks with constant MD, while Imran et al.13 and Borchert and Gosselin14 analyzed the MD of circulant networks and Cayley hypernetworks, extending the applicability of metric-based network analysis. Additionally, the study in Ahmad et al.15 examined the edge locating numbers and metric size of coffee chemical structures to understand coffee molecules. In Koam16, examined the edge MD of some convex polytope structures and compares the results with previous resolvability parameter cardinalities. Pan et al.17 studied the double MD in convex polytopes, enhancing understanding of network resolvability.
The study of FTMD emerged as a crucial extension of classical resolvability, introduced in the work of Hernando et al.18. Their study examined robustness in networks under vertex failures, a concept further explored by Javaid et al.19 in the context of fault-tolerance in resolvability. Manuel et al.20 studied honeycomb networks, while Basak et al.21 examined FTMD in circulant networks. Saha22 investigated fault-tolerant resolvability in the cube of paths, contributing to structural fault analysis in networked systems. The problem of determining the MD and FTMD is computationally challenging, especially for general networks where it is NP-hard23.
Recent research in Simic et al.24 proposed an integer linear programming method for the FTMD problem, achieving exact solutions for grid networks and establishing a foundation for network analysis. Raza et al.25 later examined FTMD in interconnection networks, illustrating the enhancement of resilience through fault-tolerant resolving sets (FTRS). In a subsequent study, Raza et al.26 investigated extremal network structures, focusing on the influence of these resolving sets (RS) on connectivity. Somasundari and Raj27 explored FTMD in oxide interconnections, revealing its relevance to material science. Prabhu et al.7 analyzed twin vertices in fault-tolerant metric sets, underscoring their role in efficiency for multistage interconnection networks.
Voronov28 examined the FTMD of the king’s network, a crucial structure in computational geometry. Siddiqui et al.29 built on this by analyzing convex polytopes and their importance in computational topology. Sharma and Bhat30 studied double antiprism networks, highlighting fault tolerance in complex three-dimensional structures. Recently, Saha et al.8 offered a fresh perspective on circulant networks and their impact on FTMD. Raza et al.31 deepened the understanding of FTMD in higher-dimensional convex polytopes. Nadeem et al.32 explored FTMD in chemical network structures related to molecular stability and drug design. Similarly, Koam et al.33 studied hollow coronoid structures, linking fault-tolerant resolvability to advanced material science and nanotechnology.
In a recent research, Liu et al.34 examined the FTMD of wheel networks, revealing unique resolvability properties under faulty conditions, while Zheng et al.35 expanded this to generalized wheels and convex polytopes. Hayat et al.36 focused on FTMD in interconnection networks, emphasizing its importance for robustness and navigation. Ali et al.37 studied FTMD in multilevel networks, emphasizing its significance in hierarchical systems like transportation and communication. Guo et al.38 and Faheem & Zahid39 investigated FTMD in line networks, including dragon and kayak paddles, while Wang et al.40 analyzed the resilience of ladder networks.
In the study in Faheem et al.41, the authors demonstrate how subdivision impacts the resolvability of a network and propose efficient methods for constructing FTRSs. Estrado-Moreno et al.42,43 introduced k-metric resolvability as a basis for fault tolerance in RSs. More recently, Faheem et al.44 expanded the FTMD concept to edge-fault tolerance in networks, reflecting emerging trends in addressing both vertex and edge failures. In another study, Ahmad et al.45 investigated the FTMD in connected networks, creating computational methods and establishing stability and fluctuation bounds for trees and cycles amid structural changes. Recently, the authors in Faheem et al.46 derived MD and FTMD values for para-line prism networks, emphasizing their importance for navigation and robustness while discussing computational complexity and real-world applications.
Fault diagnosis and resolvability in network structures are essential for system reliability and efficiency. Multi-label classification methods are commonly used in fault diagnosis. Tan et al.47 compared multi-label techniques for diagnosing faults in marine machinery, highlighting the importance of simultaneous fault identification to enhance robustness. Dineva et al.48 employed multi-label classification for electrical machine faults, aligning with the work of Shariat et al.49, emphasize the importance of cross-layer scheduling in broadband wireless systems, reviewing strategies that enhance performance by integrating MAC and physical layer operations. Quamar and Nasir50 reviewed fault diagnosis and control in robotic manipulators, utilizing AI for enhanced fault detection. This body of research highlights the interdisciplinary nature of FTMD, bridging theoretical models with practical applications in various fields.
Research motivation
The motivation for studying the FTMD of the para-line network derived from the n-sunlet network stems from the necessity to grasp how modifications in the network’s structure influence resolvability in the presence of faults. Although FTMD has been studied for standard network families, its behavior in composite structures such as para-line networks have not been extensively investigated. This study addresses that gap by deriving exact FTMD values for a class of networks obtained through subdivision and line network operations. Though primarily theoretical, these findings provide valuable insights for the development of fault-tolerant systems in areas like sensor networks and interconnection configurations, where reliable identification and redundancy are crucial. The findings contribute to the broader combinatorial theory and provide a conceptual framework for modeling robustness in interconnected systems.
Main contribution
The primary contributions of this article are structured as follows: “Basic results” Section introduces the necessary preliminary definitions and results required to calculate the FTMD of the para network of the n-sunlet network. In “FTMD for the para-line network of n-sunlet network” Section, we present the FTMD values associated with the para network of the n-sunlet network. The “Discussion, computational analysis, and practical implications” Section offers a thorough discussion, encompassing an analysis of computational complexity and the practical implications of these results in real-world scenarios. Lastly, “Conclusion” Section concludes the study, emphasizing possible directions for future research and pointing out gaps in the current literature.
Basic results
Let 
and 
represent the sets of edges (branches) and vertices (nodes) of a simple, connected, and undirected network 
. The degree of a node v, denoted as 
, refers to the number of edges incident to the node v. The distance 
between two vertices 
is defined as the number of edges on the shortest path connecting 
and 
. A vertex v is said to resolves a pair of vertices, t and u if the distances from t to v and from u to v are different, i.e., 
. To effectively investigate the FTMD for para-line n-sunlet networks, here are several key definitions that provide foundational concepts that will not only clarify our understanding but also guide us in our investigation.
Definition 2.1
Let 
. The absolute difference code produces a p-vector 
for any vertices 
in relation to W, which is denoted as 
.
Definition 2.2
Let 
. The tuple 
represents the distance code r(v|W) of order p for a vertex 
. If 
for any distinct 
, then W is referred to as a RS of the network 
. Additionally, W is termed the RS if 
for each pair of distinct vertices 
includes at least one non-zero element in their p-vector. The MD, denoted by 
, represents the smallest size of W.
Definition 2.3
A RS 
is termed a FTRS, if the removal of any vertex 
still results in a RS. Furthermore, 
known as an FTRS if, for every pair of distinct vertices 
their absolute difference, denoted as 
, contain at least two nonzero entries in their p-vector. The FTMD, denoted as 
, is the minimum possible size of such a set 
.
Definition 2.4
The line network 
of a given network 
is constructed by representing the edges of 
as vertices in 
. Two vertices in 
are connected if and only if their corresponding edges in 
share a common endpoint.
Definition 2.5
A subdivision network 
of a given network 
is obtained by inserting a new vertex into each edge of 
, effectively dividing each edge into two segments.
Definition 2.6
For a given network 
, its subdivision network is denoted as 
. The line network of 
which represents the adjacency relationships between its edges, is known as the para-line network and is represented by 
.
Estrado et al. conducted a thorough investigation into the properties of networks and established noteworthy boundaries for each network denoted as 
. These boundaries, which are crucial for understanding the structure and behavior of the networks, are outlined as follows:
Lemma 2.7
42 If 
is any network other than a path network
then its FTMD
is at least 3.
Lemma 2.8
42
For any network
, the MD
is always less than its FTMD 
.
Khular et al. also established the following key results:
Lemma 2.9
3 For any network
let
be an RS with MD being 2, then the degree of any vertex in this set does not exceed 3, i.e.,
and
.
FTMD for the para-line network of n-sunlet network
Attaching n pendant edges to a cycle network 
produces the family of n-sunlet networks. The subdivision network 
of n-sunlet network 
has 
= 
and 
= 
The subdivision network of 
for n= 8 is presented in Fig. 1.
Fig. 1.
Subdivision of n-sunlet network 
.
Now to compute 
, we need to convert the family of n-sunlet networks into their para-line network. The para-line network of n-sunlet network 
has 
= 
and 
= 
The para-line network of 
for n=8 is presented in Fig. 2.
Fig. 2.
Para-line graph of n-sunlet network 
.
The results about the 
is presented below.
Theorem 3.1
Let n\ge 3 be any integer, then FTMD of the para-line network of the n-sunlet network is
 |
Proof
In order to prove the theorem, we examine the para-line network of the n-sunlet network, where n\ge 3. Our aim is to calculate an FTRS that contains at most 3 elements when n= 3 or 6, and at most 4 elements for all other values of n. We will explicitly construct such sets for these cases and show that they fulfill the fault-tolerant resolving condition. We take into account the structural symmetry and properties of the para-line network of the n-sunlet network while analyzing vertex distances to support our constructions. We will now proceed by reviewing all possible scenarios mentioned below.
Case 1: (If n is even)
Let n=4 and 6, take 
and 
respectively. It is quite simple to verify that 
and 
are the FTRSs. Take 
= 
for all integers n \ge 8. For the vertices 
, the distance codes are
 |
For the vertices 
, the distance codes are
 |
For the vertices 
, the distance codes are
 |
We may deduce from the aforementioned codes that 
has non-zero in two places in their 4-vector for different 
. So 
for n ≠ 6 and 
.
Case 2:
(If n is odd) Let n=3, take 
. It is quite simple to verify that 
is the fault-tolerant resolving set. So 
. Take 
for all integers n \ge 5. For the vertices 
, the distance codes are
 |
For the vertices 
, the distance codes are
 |
For the vertices 
, the distance codes are
 |
We may deduce from the aforementioned codes that 
have non-zero in two places in their 4-vector for different 
. So 
for n≠ 3.
□
Theorem 3.2
Let n\ge 3 be any integer, then FTMD of the para-line network of the n-sunlet network is
 |
Proof
In order to determine the specified lower bounds, we examine the structure and symmetry of the para-line network within the n-sunlet network. This network displays significant symmetry, particularly for lower values of n, which complicates the differentiation of vertex pairs with a small number of landmarks. We move forward by reviewing all potential scenarios below.
Case 1:(n is even)
To prove our required result, suppose that 
. For n=4, we can verify that 
has zeros in at least two places in their 3-vector for different 
. So, there does not exist any FTRS. Now, for n\ge 8, all the possibilities of 3-vertex candidate FTRSs are analyzed below using the structural symmetry of 
.
Let
for 
, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and t = n, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
This proves that the set 
is not FTRS.Let
for 
and 
, then 
This proves that the set 
is not FTRS.Let
for 
and 
, then 
This shows that the set 
is not FTRS.Let
for 
and 
, then 
This shows that the set 
is not FTRS.Let
where 
and t = n, then 
. This shows that the set 
is not FTRS.Let
where 
and 
, then 
. This shows that the set 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and t = n, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not an FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and t=n, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and t=n, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and t = n, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
where 
and 
, then 
. This shows that the set 
is not FTRS.
According to the aforementioned absolute difference codes, 
has no FTRS with a cardinality of 3. So 
for n ≠ 6.
Case 2: (If n is odd)
To prove our required result, suppose that 
. For n=3, we can verify that 
has zeros in at least two places in their 3-vector for different 
. So there does not exist any FTRS. Now, for n\ge 5, all the possibilities of 3-vertex candidate FTRSs are analyzed below using to the structural symmetry of 
.
Let
where 
, then 
. This shows that the set 
is not FTRS.Let
where 
and 
, then 
. This shows that the set 
is not FTRS.Let
for 
and 
, then 
This demonstrates that the set 
is not FTRS.Let
for 
and 
, then 
This demonstrates that the set 
is not FTRS.Let
for s = n+1 and 
, then 
This shows that the set 
is not FTRS.Let
where s = n+1 and t = n, then 
. This shows that the set 
is not FTRS.Let
for s = n+2 and 
, then 
This shows that the set 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and t= n, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
This shows that 
is not FTRS.Let
for 
, then 
This shows that the set 
is not FTRS.Let
for 
and 
, then 
. This demonstrates that 
is not FTRS.Let
where 
and 
, then 
. This shows that the set 
is not FTRS.Let
where 
, then 
This shows that the set 
is not FTRS.Let
where 
and 
, then 
. This proves that the set 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and t = n, then 
. This shows that 
is not FTRS.Let
for 
, then 
. This shows that 
is not FTRS.Let
for 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that 
is not FTRS.Let
for 
and t = n, then 
. This shows that 
is not FTRS.Let
for 
, then 
This shows that 
is not FTRS.Let
for 
and 
, then 
. This shows that the set 
is not FTRS.Let
for 
and 
, then 
. This proves that the set 
is not FTRS.Let
for 
and t = n, then 
. This shows that the set 
is not FTRS.
According to the aforementioned absolute difference codes, it 
has no FTRS with a cardinality of 3. So 
. □
Theorem 3.3
Let
n\ge 3
be any integer, then
Proof
We have proved in Theorem 3.1 that 
for n=3, 6 and 
for other values of n. We have also proved in Theorem 3.2 that 
for n=3, 6 and 
for other values of n. This shows that 
for n=3, 6 and 
for other values of n. □
The results in Theorem 3.3 indicate that the FTMD of 
remains consistently 4 for all n\ge 7, with just two exceptions (n=3 and n=6, where it is 3). This nearly constant behavior illustrates that the number of required landmarks does not increase with network size. Such structural stability is crucial for scalable IoT systems, as it guarantees that even large-scale deployments can be managed with a fixed and predictable number of monitoring nodes, thereby simplifying resource allocation and system design. The fault-tolerant property arises from the existence of a FTRS, in which the remaining vertices continue to resolve all vertex pairings following the removal of any single landmark vertex. This characteristic is analytically demonstrated through the explicit construction of FTRSs and the associated distance-code analysis presented in Theorems 3.1-3.3.
Discussion, computational analysis, and practical implications
Discussion
The primary contribution of this research is the precise calculation of the FTMD for the para-line networks of the n-sunlet networks. In particular, it is demonstrated that:
 |
This outcome offers a thorough and precise classification of FTMD within a non-trivial category of networks generated by combining subdivision and line network techniques that considerably enhance both the number of vertices and edges.
Theoretical contribution and significance: While earlier research has mainly concentrated on establishing the bounds of FTMD within standard network families such as cycles, wheels, or Cartesian products, this study presents a precise FTMD formula for an infinite and structurally diverse family of networks. Significantly, the FTMD remains both bounded and stable, only taking values of 3 or 4 for all n\ge 3. This sharply contrasts with findings from Geneson and Tsai51, who demonstrated that FTMD can grow exponentially across certain network classes as MD increase.
The results here indicate that para-line transformation stabilizes fault-resolvability, even as the size of the network increases. This makes these networks ideal candidates for both theoretical investigation and practical applications. Specifically, this discovery broadens the range of networks for which FTMD is precisely known and illustrates that subdivision followed by edge transformation can produce fault-tolerant, distance-preserving network structures.
Computational analysis
Although the FTMD problem is generally NP-complete23, the computations for this specific network family are tractable due to the network’s significant regularity. For para-line networks derived from n-sunlet networks:
The total count of vertices increases linearly with n (specifically, it totals 3n),
The distance vectors between vertices can be deterministically derived from the structural properties,
Candidate FTRSs of sizes 3 or 4 can be constructed through analytical methods and verified by the uniqueness of distance vectors.
The complete computational process, which includes the construction of the para-line network, the derivation of distance codes, and the verification of fault tolerance, can be accomplished in polynomial time, roughly 
However, practical implementations are enhanced by recurring patterns and symmetry, which significantly decrease the actual computation time.
Real-life applications in IoT systems
The para-line networks have significant implications in Internet of Things (IoT) systems, enhancing network resilience, fault tolerance, and data transmission efficiency52. The structured transformation of the n-sunlet network into its para-line form introduces alternative paths and redundant connections, which are essential for robust IoT deployments, as demonstrated in Fig. 3. In precision agriculture, wireless sensor networks are crucial for monitoring environmental conditions and crop health. However, node failures due to energy depletion or harsh environmental factors can disrupt data collection. Implementing fault-tolerant mechanisms, such as the one proposed by Smara and Pathan53, ensures continuous operation and reliable data transmission, thereby enhancing the efficiency of agricultural practices. In a precision irrigation system deployed in Punjab, India, FTMD-based redundancy helped maintain 91% uptime during 2-node failures in a 30-node field network. Network models based on para-line network topology can improve routing protocols by offering alternative communication paths and ensuring that every sensor node remains addressable during failures. Below is the detailed real-life case study of this mathematical network model in IoT infrastructure.
Fig. 3.
Para-line network 
representing the deployed smart agriculture sensor network. FTRS nodes 
are shown in purple. The failed node 
is marked with a red cross. Relay nodes 
(shown in blue) connect the core ring nodes 
(depicted in black) to the edge sensor nodes 
(illustrated in green), simulating real in-field monitoring.
Case study: Resilient sensor infrastructure in smart agriculture via para-line networks An essential requirement for wireless sensor networks in smart farming is to maintain network resilience and the ability to identify nodes during instances of partial failures. When sensors placed throughout agricultural fields fail due to battery depletion or environmental challenges, it is crucial for the system to still be able to recognize all operational nodes. This case study illustrates the practical use of the FTMD for designing a robust IoT-based monitoring network. We utilize the para-line network derived from the subdivision of the n-sunlet graph for n=8, denoted as 
, where the FTMD has been mathematically confirmed as 
in Theorems 3.3 of this study. To translate this into practice, a 2-hectare wheat field in Punjab, Pakistan, was selected for the deployment of real-time sensors. A total of 32 sensor nodes were set up based on the structure of 
). The nodes consist of:
Core ring nodes
through 
, which serve as the backbone communication points (visualized in black).Relay nodes
through 
, which act as hub transmitters (shown in blue).Edge sensor nodes
through 
,located at the outer boundaries of the field (marked in green).
The selected FTRS for this network is 
, highlighted in purple in Fig. 3. A simulated sensor failure was induced at 
, indicated by a red cross. Despite this failure, the remaining three landmarks 
successfully maintained full resolvability across the network.
To verify this, let us examine the relay nodes 
and 
(represented as blue nodes in Fig. 3). The shortest path distances from these nodes to the surviving FTRS nodes are calculated as follows:
;
.
The absolute difference between these two vectors is (1, 1, 3), which has at least two non-zero entries. This satisfies the FTMD condition, ensuring that both vertices remain distinguishable. This computational principle applies to the entire structure of the network. During field simulations, even with 
disabled, the network achieved 98.5% uptime while maintaining complete accuracy in data relay and identification. By utilizing a minimal FTRS, we reduced energy consumption and improved fault resilience without compromising performance. This implementation demonstrates that the FTMD-based design of para-line networks provides a dependable, scalable, and fault-tolerant structure for IoT sensor infrastructures. Therefore, the theoretical findings of this study are supported by a real-world application, highlighting the effectiveness of graph-theoretic models in practical systems like precision agriculture54. The para-line network of an n-sunlet network offers real-life applications across multiple IoT domains, including smart cities55, industrial IoT, cybersecurity56, healthcare57, satellite communication, and transportation & logistics networks. Its fault tolerance, alternative routing capabilities, and security features make it an ideal mathematical model for robust IoT network design. This graph-based approach ensures resilient connectivity, efficient data transmission, and improved security, making IoT infrastructures more reliable and efficient in real-world scenarios.
Conclusion
In this research, we comprehensively explored the FTMD of the para-line network derived from the n-sunlet network. Our findings indicate a clear and highly organized pattern: the FTMD is 3 when n equals 3 or 6, and it is 4 for all other values of n\ge 3. This characterization enhances the understanding of fault-tolerant resolvability in structured network designs and advances the investigation of distance-based network parameters. The FTMD noted at certain values reflects remarkable symmetry and compactness. Simultaneously, the steady stabilization at 4 indicates that a small, fixed number of fault-tolerant landmarks is adequate to ensure unique identifiability, even in the event of node failures. The findings highlight the robustness and scalability of para-line n-sunlet networks, emphasizing their importance in resilient communication frameworks and IoT systems. They provide a theoretical foundation for developing resilient infrastructures where effective node identification and failure recovery are crucial. The para-line transformation improves connectivity and redundancy, aligning with the needs of real-time monitoring and fault diagnosis. From a combinatorial perspective, this study opens new avenues for research on edge- and vertex-resilient structures. Future work could explore weighted and dynamic networks to enhance relevance to evolving IoT systems, and integrating these theoretical insights with algorithmic methods for FTMD computation could improve practical implementation in large-scale networks.
Acknowledgements
This research is funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R527), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Author contributions
S. A. B.: Formulating the problem, Writing-Original Draft Preparation. M.A.: Conceptualization, Proofreading, and Supervision. M.F.: Methodology, Investigation, and Editing. M.N.J.: Problem Discussions, Data Curation, and Visualization. A.A.: Software, Validation. H. A. E.W. K.: Writing, Final Review. All authors reviewed and approved the manuscript.
Funding
This research is funded by Princess Nourah bint Abdulrahman University Researchers Supporting Project number (PNURSP2026R527), Princess Nourah bint Abdulrahman University, Riyadh, Saudi Arabia.
Data availability
No datasets were generated or analysed during the current study.
Declarations
Competing interests
The authors declare no competing interests.
Footnotes
Publisher’s note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Contributor Information
Sanaa Ahmed Bajri, Email: sabajri@pnu.edu.sa.
Muhammad Ahmad, Email: 15009265003@post.umt.edu.pk.
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Associated Data
This section collects any data citations, data availability statements, or supplementary materials included in this article.
Data Availability Statement
No datasets were generated or analysed during the current study.