Abstract
Fault tolerance is the characteristic of a system that permits it to carry on its intended operations in case of the failure of one of its units. Such a system is known as the fault-tolerant self-stable system. In graph theory, if we remove any vertex in a resolving set, then the resulting set is also a resolving set, called the fault-tolerant resolving set, and its minimum cardinality is called the fault-tolerant metric dimension. In this paper, we determine the fault-tolerant resolvability in line graphs. As a main result, we computed the fault-tolerant metric dimension of line graphs of necklace and prism graphs (2010 Mathematics Subject Classification: 05C78).
1. Introduction and Preliminaries
Let be a simple connected graph with vertex set and edge set . The distance between two vertices is the length of a shortest path between them. The degree of a vertex is the number of edges that are incident to it. Let be an ordered set and ; then, the representation of with respect to is the -tuple . is called a resolving set if different vertices of have different representations with respect to . A resolving set with a minimum number of elements is called a basis for , and the cardinality of the basis is known as the metric dimension of , represented by . For , the th component of is 0 if and only if . Hence, to prove that is a resolving set, it is enough to show that for each pair . The absolute difference representation of with respect to is . So, is a resolving set if has at least one entry in the -vector different from zero for .
The concept of metric dimension of the general metric space was presented in 1953 (see [1]). After twenty years, the concept of resolving set in graphs was first introduced by Slater [2, 3] in 1975 and also independently by Harary and Melter [4] in 1976. The resolving sets were basically defined to determine the location of the intruder in a network, but later, Chartrand and Zhang used metric bases in the fields of robotics, chemistry, and biology in 2003, see [5, 6].
It will be a difficult task to locate an interrupter if one of the sensors does not function in a proper way. In order to tackle such problems, Hernando et al. [7] gave the idea of the fault-tolerant resolving set. Fault-tolerant resolving set is a resolving set if the removal of any element keeps it resolving. Formally, a resolving set of any graph is called a fault-tolerant resolving set if for all is also a resolving set of . The minimum cardinality of the fault-tolerant resolving set is called the fault-tolerant metric dimension, and it is denoted by . In other words, for all , has at least two entries in the -vector different from zero.
Fault-tolerant metric dimension is an interesting concept and has been studied by many authors. For instance, Hernando et al. in [7] computed the fault-tolerant resolving set for tree graphs and proved that for the path graph on vertices. Voronov in [8] computed the fault-tolerant metric dimension of the king’s graph. Recently, Hussain et al. in [9] computed closed formulas for the fault-tolerant metric dimension of wheel-related graphs. Raza et al. in [10] computed the fault-tolerant metric dimension of some classes of convex polytopes. Raza et al. in [11] showed that the fault-tolerant metric dimension of the complete graph on vertices is . Javaid et al. in [12] proved that for the cycle graph on vertices. For more details on the fault-tolerant metric dimension, see [13–15].
The line graph of graph is the graph whose vertices are the edges of , and two vertices and of are connected if and only if they have a common end vertex in . The metric dimension of line graphs is studied in [6, 10, 16–18]. For more details, see [19–21]. Here, we determine the fault-tolerant metric dimension in line graphs. The fault-tolerant metric dimension in line graphs is only known for path and cycle graphs as given in the following theorem.
Theorem 1. The fault-tolerant metric dimension of the line graphs of path and cycle graphs of order is 2 and 3, respectively.
Proof. The results are obvious from the definition of the line graph, and the results are proved in [7, 12], respectively. Since it is difficult to compute the exact values of for every graph , A. Estrado-Moreno et al. gave some of the important bounds on the fault-tolerant metric dimension of graphs as follows.
Lemma 1 (see [22]). Let be any graph; then, .
Lemma 2 (see [22]). If , then for all .
In [23], Khuller et al. studied an important property of graphs with metric dimension 2 as follows.
Lemma 3 (see [23]). Let be a graph with metric dimension 2, and let be a resolving set in . Then, the degree of both and is at most 3.
Consequently, similar argument works for the graphs with fault-tolerant metric dimension 3 are given in the following lemma.
Lemma 4. Let be a graph with fault-tolerant metric dimension 3, and let be a fault-tolerant resolving set in . Then, the degree of each vertex , , and is at most 3. The rest of the paper is structured as follows: in Section 2, we will compute the fault-tolerant metric dimension of the line graph of the necklace graph. In Section 3, we will compute the fault-tolerant metric dimension of the line graph of the prism graph.
2. The Fault-Tolerant Metric Dimension of the Line Graph of the Necklace Graph
The necklace graph for consists of the edge set as shown in Figure 1.
For the fault-tolerant metric dimension of the line graph of the necklace graph, we have to construct a line graph of with (see Figure 2).
In the following theorem, the result for the metric dimension of the line graph of the necklace graph is given.
Theorem 2 (see [17]). The metric dimension of the line graph of the necklace graph is 3 for . Now, we will compute the fault-tolerant metric dimension of the line graph of the necklace graph.
Theorem 3. The fault-tolerant metric dimension of the line graph of the necklace graph is 4 for .
Proof. To prove this theorem, consider the following cases.
Case 1 ( is odd). Let , and take . Representation of vertices with respect to isNow, representation of vertices with respect to isRepresentation of vertices with respect to isIt can be easily seen that has at least two entries in the 4-vector different from zero for any . Hence, is a fault-tolerant resolving set of . So, by using Theorem 2 and Lemma 1, for every odd .
Case 2 ( is even). Let , and take . For , it is easy to verify that all the representations are distinct.
Now, representation of vertices for with respect to isRepresentation of vertices for with respect to isRepresentation of vertices for with respect to isIt can be easily seen that has at least two entries in the 4-vector different from zero for any . Hence, is a fault-tolerant resolving set of . So, by using Theorem 2 and Lemma 1, for every even .
3. The Fault-Tolerant Metric Dimension of the Line Graph 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 edge set as shown in Figure 3. The line graph of the prism graph consists of 3-sided faces, 4-sided faces, and -sided faces as shown in Figure 4. For our purpose, we label the inner cycle vertices of by , middle vertices by , and the outer cycle vertices by .
In the following theorem, the result for the metric dimension of the line graph of the prism graph is presented.
Theorem 4 (see [24]). Let be the prism graph; then, the metric dimension of the line graph of the prism graph is 3 for . Now, in the following theorem, we will compute the fault-tolerant metric dimension of the line graph of the prism graph.
Theorem 5. Let be the prism graph; then, the fault-tolerant metric dimension of the line graph of the prism graph is 4 for .
Proof. To prove this theorem, consider the following cases.
Case 1 ( is even). Let , and take . Representation of vertices with respect to isRepresentation of vertices with respect to isRepresentation of vertices when with respect to isRepresentation of vertices when with respect to isIt can be easily seen that has at least two entries in the 4-vector different from zero for any . Hence, is a fault-tolerant resolving set of . So, by using Theorem 4 and Lemma 1, for every even .
Case 2 ( is odd). For , take . It is easy to check that is a fault-tolerant resolving set. Let , and take . Representation of vertices with respect to isRepresentation of vertices with respect to isRepresentation of vertices when with respect to isRepresentation of vertices when with respect to isIt can be easily seen that has at least two entries in the 4-vector different from zero for any . Hence, is a fault-tolerant resolving set of . So, by Theorem 4 and Lemma 1, for every odd .
4. Conclusion
In this paper, we have studied for the first time the fault-tolerant metric dimension of the line graph of a graph. We have given the exact values of the fault-tolerant metric dimension of the line graphs of necklace and prism graphs and found these values are independent of the number of vertices of a graph. In future, we will compute the exact value of the fault-tolerant metric dimension of the line graph of the kayak paddle graph. We will also discuss the fault-tolerant resolvability in graphs by using other graph operations.
Data Availability
The data used to support the findings of this study are included within the article.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Authors’ Contributions
All authors contributed equally to this study.
Acknowledgments
This paper was supported by the Educational Commission of Pakistan.