Fault-Tolerant Resolvability in Some Classes of Subdivision Graphs

Faheem, Muhammad; Zahid, Zohaib; Alrowaili, Dalal; Siddique, Imran; Iampan, Aiyared

doi:https://doi.org/10.1155/2022/5784800

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Distances in Graph Theory

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Research Article | Open Access

Volume 2022 | Article ID 5784800 | https://doi.org/10.1155/2022/5784800

Fault-Tolerant Resolvability in Some Classes of Subdivision Graphs

Muhammad Faheem,¹Zohaib Zahid,¹Dalal Alrowaili,²Imran Siddique,¹and Aiyared Iampan³

Academic Editor: Gul Rahmat

Received09 Nov 2021

Accepted13 Jan 2022

Published18 Feb 2022

Abstract

The concept of resolving sets (RSs) and metric dimension (MD) invariants have a wide range of applications in robot navigation, computer networks, and chemical structure. RS has been used as a sensor in an indoor positioning system to find an interrupter. Many terminologies in machine learning have also been used to diagnose the interrupter in the systems of marine and gas turbines using sensory data. We proposed a fault-tolerant self-stable system that allows for the detection of an interrupter even if one of the sensors in the chain fails. If the elimination of any element from a RS is still a RS, then the RS is considered as a fault-tolerant resolving set (FTRS), and the fault-tolerant metric dimension (FTMD) is its minimum cardinality. In this paper, we calculated the FTMD of the subdivision graphs of the necklace and prism graphs. We also found that this invariant has constant values for both graphs.

1. Introduction and Preliminaries

Let be a connected, undirected, finite, and simple graph with vertex set and edge set . The number of edges in the shortest path between two vertices is known as the distance between them. The cardinality of edges that are incident to a vertex is called degree of . Let be an ordered subset of and ; then, the -tuple is the distance-vector representation of with respect to and is denoted by . If distance-vector representations are distinct for every vertex of , then is known as the RS. The metric basis of is the minimum number of elements in a RS, and the cardinality of the basis is called the MD of , denoted by . For , the th component of is 0 if and only if . So, would be a RS if for each pair . The absolute difference representation consists of -vector for any with respect to , and it is denoted by . The absolute difference representation is another concept to define the RS. If has at least one nonzero element in its -vector for every , then is called the RS. For more details, we refer to [1, 2].

In 1975, when Slater was working with Coast Guard Loran and US Sonar (long range aids to navigation) station, he described the notion of RSs in graphs [3, 4]. Harary and Melter independently described this idea in 1976 [5]. The RSs were introduced to locate the interrupter in a computer network. After that, in 2003, Chartrand and Zhang used metric bases in the fields of biology, robotics, and chemistry [6, 7]. Many techniques have been used in machine learning to find the interrupter using sensory data. In 2021, Tan et al. compared the data-driven and model-driven methods for the diagnosis of interrupters in marine machinery systems [8].

It will be a difficult task to locate an interrupter if one of the sensors does not work properly. To tackle such problems, Hernando et al. gave the idea of a FTRS [9]. If the elimination of any element from a RS is again a RS, then the RS is considered as a FTRS. Formally, for any RS of , if is also a RS for any , then is called a FTRS of . In other words, contains at least two elements that are nonzero in the -vector for all . The FTMD is the minimum cardinality of a FTRS, and it is denoted by .

Hernando et al. in [9] studied the concept of the FTMD and computed it for the tree graph. They also found an important upper bound for the FTMD of graph , which is . Voronov in [10] determined the upper bound for the FTMD of the king’s graph. Hussain et al. in [11] proved that the FTMD is constant for the antiweb graph and unbounded for the gear and antiweb gear graphs.

Raza et al. in [12] studied the bounds for the FTMD of some families of convex polytopes. In [13], Raza et al. proved some bounds for the FTMD of extended Petersen, antiprism, and squared cycle graphs. Raza et al. in [14] found some bounds for the FTMD by considering the graphs of -dimension grid networks, hexagonal networks, and honeycomb networks. Zheng et al. in [15] calculated the precise values for the FTMD of the generalized wheels and some families of convex polytopes. Afzal and others in [16] gave some important results and proved that there exist some families of convex polytopes, which have unbounded MD and FTMD. Basak et al. in [17] calculated the FTMD for the circulant graph. Somasundari and Raj in [18] calculated the fault-tolerant resolvability for the interconnections of oxide networks.

Javaid et al. in [19] computed the exact value of the FTMD of the cycle graph. Hayat et al. in [20] calculated some upper bounds for the FTMD for -dimensional bens, -dimensional butterfly, and silicate networks. Simic et al. in [21] contributed great work about the FTMD and computed the precise value for the grid graph. Laxman in [22] computed the lower bound of the FTMD for the cube of the path graph. Recently, the FTMD for the line graphs was studied by Guo et al. in [23], and they computed it for the line graphs of the prism and necklace graphs.

The subdivision graph of any graph can be obtained by adding a new vertex to each edge of the graph as shown in Figures 1 and 2. In this paper, we computed the FTMD of the subdivision graphs of the necklace and the prism graphs. The results of the FTMD in subdivision graphs are known only for cycle and path graphs as shown in the following theorem.

Theorem 1. For any and , we have and , respectively, where is the path graph and is the cycle graph.

Since it is not an easy task to calculate for any graph , Estrado-Moreno et al. computed some important bounds on of every graph as follows.

Lemma 1 (see [24]). Let be any graph; then, we have .

Lemma 2 (see [24]). Let be any graph; then, we have .

Khuller et al. [25] gave an essential result for every graph with 2 of its MD as follows.

Lemma 3 (see [25]). Let , for any graph , and let be a RS in . Then, and .

Therefore, similar property works for every graph with 3 of its as presented in the following lemma.

Lemma 4. Let , for any graph , and let be a FTRS in . Then, , , and .

We will discuss and compute the precise results of as well as , where is a necklace graph in Section 2. We will calculate the exact value of by considering as a prism graph in Section 3.

2. The Fault-Tolerant Metric Dimension for the Subdivision of the Necklace Graph

The necklace graph for consists of the vertices as shown in Figure 3.

Now, to compute , we need to convert the necklace graph into its subdivision graph. The subdivision graph of the necklace graph is shown in Figure 1.

The result about the MD for the subdivision graph of the necklace graph is presented in the following theorem.

Theorem 2. For any integer , we have .

Proof. Following are the cases to compute the required results:

Case 1. (when is odd).
Take for all odd integers . We need to prove that is the RS for the graph . To prove this, we give distance-vector representations of every vertex in . Representation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isThe distance-vector representations are distinct for all the vertices of . So, is the RS. Hence, . Now, in order to prove that , suppose contrary that , and according to Lemma 3, we have the following possibilities:(1)Let for ; then, we have the following subcases:(i) for .(ii) for . So, is not the RS.(2)Let for ; then, we have the following subcases:(i) for .(ii) for . So, is not the RS.(3)Let for and ; then, we have the following subcases: (a) When :(i) for .(ii) for and .(iii) for and . (b) When :(i) for .(ii) for and .(iii) for and . So, is not the RS.(4)Let for and ; then, we have the following subcases: (a) When :(i) for and .(ii) for and .(iii) for and .(iv) for and . (b) When :(i) for .(ii) for and .(iii) for and .(iv) for and .(v) for and .(vi) for and .(vii) for and . (c) When :(i) for .(ii) for and .(iii) for and .(iv) for and .(v) for and . So, is not the RS.(5)Let for and ; then, . So, is not the RS.(6)Let for and ; then, we have the following subcases:(i) for .(ii) for . So, is not the RS.(7)Let for and ; then, we have the following subcases:(i) for .(ii) for . So, is not the RS.(8)Let for and ; then, we have the following subcases:(i) for and .(ii) for and .(iii) for and .(iv) for and .(v) for and . So, is not the RS.(9)Let for and ; then, we have the following subcases: (a) When :(i) for and .(ii) for and .(iii) for and .(iv) for and .(10)Let for and ; then, we have the following subcases:(i) for and .(ii) for and .(iii) for and .(iv) for and .(v) for and . So, is not the RS.We deduce to contradict in all the possibilities. So, there is no RS with cardinality 2. This shows that . Hence, .

Case 2. (when is even).
Take for all even integers . Representation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isThe distance-vector representations are distinct for all the vertices of . So, is the RS. Hence, . Now, in order to prove that , suppose contrary that , and according to Lemma 3, we have the following possibilities:(1)Let for ; then, . So, is not the RS.(2)Let for ; then, . So, is not the RS.(3)Let for and ; then, we have the following subcases: (a) When :(i) for .(ii) for and .(iii) for and .(b)When :(i) for .(ii) for and .(iii) for and . So, is not the RS.(4)Let for ; then, we have the following subcases: (a) When :(i) for .(ii) for and .(iii) for and .(iv) for and .(v) for and . (b) When :(i) for .(ii) for and .(iii) for and .(iv) for and .(v) for and .(vi) for and . (c) When :(i) for .(ii) for and .(iii) for and .(iv) for and .(v) for and . So, is not the RS.(5)Let for and ; then, . So, is not the RS.(6)Let for and ; then, we have the following subcases:(i) for .(ii) for and .(iii) for and . So, is not the RS.(7)Let for and ; then, we have the following subcases:(i) for .(ii) for . So, is not the RS.(8)Let for and ; then, we have the following subcases:(i) for and .(ii) for and .(iii) for and .(iv) for and .(v) for and . So, is not the RS.(9)Let for and ; then, we have the following subcases:(i) for and .(ii) for and .(iii) for and .(iv) for and .(v) for and .(vi) for and . So, is not the RS.(10)Let for and ; then, we have the following subcases:(i) for and .(ii) for and .(iii) for and .(iv) for and . So, is not the RS.We deduce to contradict in all the possibilities. So, there is no RS with cardinality 2. This shows that . Hence, .

Now, the following theorem presented the exact value of the FTMD of the subdivision graph of the necklace graph.

Theorem 3. For any integer , we have .

Proof. Following are the cases to calculate required results:

Case 3. (when is odd).
Take for all odd integers . We need to prove that is the FTRS for the graph . To prove this, we give distance-vector representations of every vertex in . Representation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isThe distance-vector representations are distinct for all the vertices of . So, is the RS. We can also verify that, for every , at least two elements in the 4-vector are nonzero. So, the RS becomes a FTRS. This shows that . By using Lemma 1 and Theorem 2, we have . Hence, .

Case 4. (when is even).
Take for all even integers . Representation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices for where isRepresentation of the vertices for where isRepresentation of the vertices where isThe distance-vector representations are distinct for all the vertices of . So, is the RS. We can also verify that, for every , at least two elements in the 4-vector are nonzero. So, the RS becomes a FTRS. This shows that . By using Lemma 1 and Theorem 2, we have . Hence, .

3. The Fault-Tolerant Metric Dimension for the Subdivision of the Prism Graph

The prism graph is Cartesian product graph , where is the cycle graph of order and is a path of order 2. The prism graph consists of 4-sided faces and -sided faces with vertex set as shown in Figure 4.

Now, to compute , we need to convert the prism graph into its subdivision graph. The subdivision graph of the prism graph is shown in Figure 2.

The known result of the MD of the subdivision graph of the prism graph is given in the following theorem.

Theorem 4 (see [26]). For any integer , we have .

Here, we will find the exact value of , which is presented in the following theorem.

Theorem 5. For any integer , we have .

Proof. Following are the cases to compute required results:

Case 5. (when is even).
Take for all even integers . We need to prove that is the FTRS for the graph . To prove this, we give distance-vector representations of every vertex in . Representation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isThe distance-vector representations are distinct for all the vertices of . So, is the RS. We can also verify that, for every , at least two elements in the 4-vector are nonzero. So, the RS becomes a FTRS. This shows that . By using Lemma 1 and Theorem 4, we have . Hence, .

Case 6. (when is odd).
For , take . It is quite simple to check that is a FTRS.
Take for all odd integers . Representation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isRepresentation of the vertices where isThe distance-vector representations are distinct for all the vertices of . So, is the RS. We can also verify that, for every , at least two elements in the 4-vector are nonzero. So, the RS becomes a FTRS. This shows that . By using Lemma 1 and Theorem 4, we have . Hence, .

4. Conclusion

In this paper, we have studied, for the first time, the FTMD of the subdivision graph of a graph. We have found that the FTMD of the subdivision graphs of the necklace and prism graphs is exactly greater than one of their metric dimensions.

Data Availability

No data were used in this manuscript.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Muhammad Faheem et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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