1. Introduction and Preliminaries
The notion of metric dimension originated in the twentieth century by the work done by Slater [1, 2] and latter independently by Harary and Melter [2]. The initial concept was introduced to locate the intruder in any network but the development and impact of the notion was far reaching in coming years. Several applications of metric related network parameters can be seen in social networking, navigation, communications, and engineering, pharmaceutical chemistry. The fault tolerance of a networking system is its ability to perform in case of malfunction of one of the source nodes. The fault tolerant metric dimension of a network can be regarded as a process of uniquely identification of each node of a network in case a malfunction occurs in one of the source node. This allows fault tolerant metric dimension more adaptability and flexibility for practical purposes than the parent notion.
For a simple network , the size of least distant route between any two of its nodes is referred as distance between them, notated as . If is an arrayed collection of nodes in then the vector is called a representative vector of the node wrt. notated as . The collection containing distinct representative vectors corresponding to distinct nodes is referred as resolving collection of nodes. A resolving collection of nodes with least members is called basis for the network and its size is referred as metric dimension of , represented by .
The limitations in the parent notion of metric dimension of having deviation at only one position between the representative tuples of vertices led Estrado-Moreno et al., to introduce a generalized notion of metric dimension in 2014 (see [3]). Thus, the notion of the metric dimension was introduced, which becomes the metric dimension, for , denoted by . The equivalent condition on the existence of metric basis and related results were presented by Estrado-Moreno et al. in [4]. Further the notion was explored in context of generalized metric spaces in [6] and respectively for lexicographic product of graphs and corona product of graphs, in [5, 6]. The extension of this concept to the more general case of non-necessarily connected graphs is studied in [7]. The complexity of some metric dimension problems revealed that its computation is NP-hard (see [8]). This motivated the study of notion metric dimension problems for particular values of . The notion for , is referred as fault tolerant metric dimension, which was introduced by Hernando in 2008 (see [9]). Here, we include a formal definition of the notion: Let be an arrayed collection of nodes in . If for each pair of nodes their absolute difference representation contains more than one zeros, then the collection is referred as a fault-tolerant resolving set (FTRS) for . An FTRS of the least size in is called as fault tolerant metric basis (FTMB) and its size is the fault tolerant metric dimension (FTMD) of , notated by .
The notion FTMD have been extensively discussed by several researchers. In this regard, FTMD for prism related graphs and circulant graphs have been studies in [10, 11], FTMR of lexicographic product and some other classes have been discussed in [12, 13], FTMD of certain wheel related graphs can be seen in [14]. Further, FTMD for convex polytopes and triangular lattices are computed in [15, 16]. Recently, in 2020, Huo et al. computed FTMD for generalized prism graph and Mobious ladders and latter in 2021 Bashir et al. discussed FTMD of some classes of rotationally symmetric graphs (see [17, 18]). Some other related developments can also be seen in [19, 20]. Following are the two theorems which will be helpful in computing our main results.
Theorem 1 (see [3]). For any graph.
Theorem 2 (see [3]). If, then.
The symmetric planer graphs, like generalized Petersen and sunlet networks have key importance in the fields of telecommunication, navigation and networking due to the structure of these networks which results in uniform rate of data transfer and thereby optimizing the resources used.
1.1. Main Results
The study conducted in this article, lead to following main results
Theorem 3. (1)For ,(2)For ,(3)For ,
The rest of the article is organized in the following manner: In Section 2, the FTMD of family of sunlet graph is computed. The Section 3 comprises of the computation of FTMD of family the generalized Petersen graph , for . We also computed the FTMD of for even and some tight bounds are obtained for odd . An application of the current work in context of navigational routing problem is furnished in Section 4. Lastly, the paper is concluded with open problems in Section 5.
2. The FTMD of family of the sunlet graph
The family of sunlet graph denoted by is the graph obtained by attaching pendant edges to a cycle graph as shown in Figure 1. The vertex set and edge set , where subscripts are to be read modulo . In the following lemma, the metric dimension of the family of sunlet graph is presented.
Lemma 1. The metric dimension of , for is
Proof. In order to prove the theorem, following cases can be considered:
Case 1. (When is odd)
Let , for and take . Representation of the vertices and with respect to is shown in Table 1.
We can see that for all , . Hence, is a resolving set. This implies that . Since, if and only if is a path graph, therefore, .
Case 2. (When )
It can be easily verified that is a minimal resolving set for . Therefore, we have .
Case 3. (When is even and )
Let , for and take . Representation of the vertices and with respect to is shown in Table 2.
We can see that for all , . Hence, is a resolving set. Therefore, . It is exclusively required to prove that , for . This can be achieved by showing that is unable to have a resolving set of order 2, leading to the following possibilities:a)If with , then .b)If , then .c)If , then .d)If , then .Hence, in all above possibilities, it is concluded that does not have a 2 cardinality resolving set. This concludes the proof. □
The above lemma will be helpful in the following result.
Theorem 4. The FTMD of , for is
Proof. In order to show the assertion, following cases can be considered:
Case 1.
It can be immediately confirmed that , when and , when , are FTRSs for . This implies that . Since, by Lemma 1, has metric dimension 2 (when is odd), therefore, by using Theorem 1, it is clear that . This implies that .
Case 2. (When )
Take . Then the representation of the vertices and with reference to the above said set are . and .
We can observe that more than one zeros exist in the , for each . Hence, is a FTRS. Therefore, in view of Lemma 1 togather with Theorem 1, we conclude that . The only thing that remains to show is . The Table 3 shows that has no FTRS of cardinality 3. Hence, .
Case 3. (When is even and )
Let , for and take . Representation of the vertices and with respect to is shown in Table 4.
We can observe that more than one zeros exist in the , for each . Hence, is a FTRS. Now, by Lemma 1 and Theorem 1, we have .
Case 4. (When is odd and )
Let with and take . Representation of the vertices and with respect to is shown in Table 5.
Therefore, in lights of Lemma 1 combined with the Theorem 1, we have . The only thing that remains to show is . In order to achieved that is unable to have a FTRS of order 3, we have the following possibilities:a)If such that and , then .b)If such that and , thenFor pendant vertices , consider with and . The proofs are similar as Cases and respectively.c)If , such that and , then for , andd)If , such that and , thenand if , with , then .e)If , such that and , thenand for .f)If , such that and , thenHence, in all above possibilities, we conclude that there is no FTRS for containing exactly 3 nodes. This concludes the proof. □
3. The FTMD of family of The Generalized Petersen graphs
Coxeter was the first one to introduce the generalized notion of Petersen graphs in 1950 (see [23]). It is an important class of graphs with vertex set and edge set , where subscripts are to be read modulo and . The Petersen graphs and are shown in Figures 2 and 3 respectively.
The forthcoming lemma will be resourceful in the computation the FTMD in regards to family of the generalized Petersen graph .
Lemma 2 (see [24]). For the generalized Petersen graph;
Theorem 5. The FTMD of, foris
Proof. In order to show the assertion, following cases can be considered:
Case 1. (When is even)
Take , then for the vertices and the representation is shown in Table 6.
Since it can be observed that more than one zeros exist in the , for each . Therefore, . Hence, in view of Lemma 2 together with the Theorem 1, we conclude that .
Case 2. (When is odd)
This case is further subdivided as follows:
Case 2a. (When )
For is can be confirmed immediately that is its FTRS. This implies that . Therefore, in view of the Lemma 2 and the Theorem 1, we conclude that .
Case 2b. (When is odd and )
Let , for and take , then for the vertices and the representation is shown in Table 7.
Since it can be observed that more than one zeros exist in the , for each . Therefore, . Hence, in view of Lemma 3 and Theorem 1, we have . It is exclusively required to prove that . This can be achieved by showing that is unable to have a resolving set of order 3, leading to the following possibilities:a)If such that and , then .b)If such that and , thenc)If , such that and , then and if , thend)If , such that and , thenand if and then .
Therefore, all the aforesaid cases reveal that have not FTRS of cardinality 3. This concludes the proof. □
The upcoming theorem will be aided by the following lemma.
Lemma 3 (see [21]). For,.
Theorem 6. The FTMD of, foris
Proof. In order to prove the theorem, following cases can be considered:
Case 1. (When is even)
The case is further subdivied as follows:
Case 1a. (When )
It can be immediately confirmed that is a FTRS for . This implies that . Therefore, in lights of Lemma 3.3 combined with Theorem 1, it is concluded that .
Case 1b. (When and )
Let , for even and take . Then for the vertices and the representations are shown in Table 8 and 9.
Case 1c. (When and )
Let , for odd and take . Then for the vertices and the representations are shown in Table 10 and 11.
The Cases 1b and 1c indicates that more than one zeros exist in the , for each implying that . Therefore, in lights of Lemma 3 combined with the Theorem 1, we conclude that .
Case 2. (When mod 2)
The case is further divided as below:
Case 2a. (When )
It can be immediately confirmed that and is FTRS for and respectively. This implies that for . Therefore, in lights of Lemma 3 combined with Theorem 1, it is concluded that for .
Case 2b. (When and )
Let , for even and take , then for the vertices and the representations are shown in Table 12 and 13.
Case 2c. (When and )
Let , for odd and take ,then the representations of the vertices and are shown in Table 14 and 15.
The Cases 2b and 2c indicates that more than one zeros exist in the , for each . Hence, is a FTRS. Hence, in view of Lemma 3 combined with the Theorem 1, we conclude that . This concludes the proof. □
Remark 1. This can be verified easily that , when , by using the resolving set . Similarly, it can be verified that , for .
4. Application
The current section is included with an application of FTMD in context of navigational optimization problem is discussed. The essence of a navigational routing problem is to reach targeted location and avoiding mixing of apparently similar locations. If the targeted locations are referred as nodes and the roads connecting them as edges of a network, then the process of uniquely identifying each node of the network with respect to certain minimum collection of its reference nodes is a realization of metric dimension in networks. Further, if one of the location in the reference collection of nodes is unreachable, then the minimum number of such reference nodes providing unique identification of each location in network is realization of FTMD of network. As an illustrative case, consider navigating in a network , consisting of node set as shown in Figure 3. Then the unique identification of each node with respect to minimum collection of reference nodes is shown in Table 16.
It can be seen that the set is minimum set representing each node uniquely even if one of the nodes in it is unreachable. Therefore, it is concluded that if the navigation coordinates is in accordance with the set , then the navigational routing will be optimal.
5. Conclusion
In the current study, the FTMD of the families sunlet graph and the generalized Petersen graph , for is computed, which were found to be constant. We also computed the constant FTMD of for even and tight bounds are obtained for odd . Finally, the article is concluded with the following open problems. [22–25].
Open Problem 1. For what values of odd the FTMD of family of the generalized Petersen graph is 4 or 5.
Open Problem 2. Find the FTMD of family of the generalized Petersen graphs for .
Data Availability
All data are included within this article. However, the reader may contact the corresponding author for more details of the data.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
The authors are grateful to the reviewers for their valuable suggestions to improve the presentation of the manuscript.