Fault-Tolerant Resolvability in Some Classes of Line Graphs

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 Clas-siﬁcation: 05C78).


Introduction and Preliminaries
Let G be a simple connected graph with vertex set V(G) and edge set E(G). e distance d(s, t) between two vertices s, t ∈ V(G) is the length of a shortest path between them. e degree of a vertex s is the number of edges that are incident to it. Let W � w 1 , w 2 , . . . , w l ⊂ V(G) be an ordered set and s ∈ V(G); then, the representation r(s | W) of s with respect to W is the l-tuple (d(s, w 1 ), d(s, w 2 ), . . . , d(s, w l )). W is called a resolving set if different vertices of G have different representations with respect to W. A resolving set with a minimum number of elements is called a basis for G, and the cardinality of the basis is known as the metric dimension of G, represented by β(G). For W � w 1 , w 2 , . . . , w l ⊂ V(G), the ith component of r(s | W) is 0 if and only if s � w i . Hence, to prove that W is a resolving set, it is enough to show that r(s | W) ≠ r(t | W) for each pair s ≠ t ∈ (V(G)/W). e absolute difference representation of s, t ∈ V(G) with respect to W is A D((s, t) | W) � (|d G (s, w 1 ) − d G (t, w 1 )|, . . . , |d G (s, w l ) − d G (t, w l )|). So, W is a resolving set if A D((s, t) | W) has at least one entry in the l-vector different from zero for s ≠ t ∈ V(G). e 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. e 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 W ′ of any graph G is called a faulttolerant resolving set if (W ′ / w { }) for all w ∈ W ′ is also a resolving set of G. e minimum cardinality of the faulttolerant resolving set is called the fault-tolerant metric dimension, and it is denoted by β ′ (G). In other words, for all s, t ∈ V(G), AD((s, t) | W ′ ) has at least two entries in the l-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 β ′ (P n ) � 2 for the path graph P n on n ≥ 2 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 n vertices is n. Javaid et al. in [12] proved that β ′ (C n ) � 3 for the cycle graph C n on n ≥ 3 vertices. For more details on the fault-tolerant metric dimension, see [13][14][15]. e line graph L(G) of graph G is the graph whose vertices are the edges of G, and two vertices e and f of L(G) are connected if and only if they have a common end vertex in G.
e metric dimension of line graphs is studied in [6,10,[16][17][18]. For more details, see [19][20][21]. Here, we determine the fault-tolerant metric dimension in line graphs. e fault-tolerant metric dimension in line graphs is only known for path and cycle graphs as given in the following theorem.

Theorem 1.
e fault-tolerant metric dimension of the line graphs of path and cycle graphs of order n is 2 and 3, respectively. Proof.
e 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 β ′ (G) for every graph G, A. Estrado-Moreno et al. gave some of the important bounds on the fault-tolerant metric dimension of graphs as follows.
In [23], Khuller et al. studied an important property of graphs with metric dimension 2 as follows.
Lemma 3 (see [23]). Let G be a graph with metric dimension 2, and let v 1 , v 2 ⊂ V(G) be a resolving set in G. en, the degree of both v 1 and v 2 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 G be a graph with fault-tolerant metric dimension 3, and let
and v 3 is at most 3. e 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.

The Fault-Tolerant Metric Dimension of the
Line Graph of the Necklace Graph e necklace graph N e n for n ≥ 2 consists of the edge set E(N e n ) � f i , g i , h i : 1 ≤ i ≤ n + 1 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 L(N e n ) of N e n with n ≥ 2 (see Figure 2).
In the following theorem, the result for the metric dimension of the line graph of the necklace graph is given.
e metric dimension of the line graph of the necklace graph L(N e n ) is 3 for n ≥ 2. Now, we will compute the fault-tolerant metric dimension of the line graph of the necklace graph. Proof. To prove this theorem, consider the following cases.

Mathematical Problems in Engineering
Now, representation of vertices g i with respect to W ′ is Representation of vertices h i with respect to W ′ is h n+1

Mathematical Problems in Engineering 3
It can be easily seen that AD((s, t) | W ′ ) has at least two entries in the 4-vector different from zero for any s, t ∈ V(L(N e n )). Hence, W ′ is a fault-tolerant resolving set of L(N e n ). So, by using eorem 2 and Lemma 1, β ′ (L(N e n )) � 4 for every odd n.
Case 2 (n is even). Let n ≥ 2, and take W ′ � f (n+2/2) , h 1 , h (n+2/2) , g (n+2/2) } ⊆ V(L(N e n )). For n � 2, it is easy to verify that all the representations are distinct. Now, representation of vertices f i for n ≥ 4 with respect to W ′ is Representation of vertices g i for n ≥ 4 with respect to W ′ is Representation of vertices h i for n ≥ 4 with respect to W ′ is

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It can be easily seen that AD((s, t) | W ′ ) has at least two entries in the 4-vector different from zero for any s, t ∈ V(L(N e n )). Hence, W ′ is a fault-tolerant resolving set of L(N e n ). So, by using eorem 2 and Lemma 1, β ′ (L(N e n )) � 4 for every even n.

The Fault-Tolerant Metric Dimension of the
Line Graph of the Prism Graph e prism graph Y n is Cartesian product graph C n × P 2 , where C n is the cycle graph of order n and P 2 is a path of order 2. e prism graph Y n consists of 4-sided faces and n-sided faces with edge set E(Y n ) � e i , f i , g i ; 1 ≤ i ≤ n as shown in Figure 3. e line graph L(Y n ) of the prism graph consists of 3-sided faces, 4-sided faces, and n-sided faces as shown in Figure 4. For our purpose, we label the inner cycle vertices of L(Y n ) by e i : 1 ≤ i ≤ n , middle vertices by f i : 1 ≤ i ≤ n , and the outer cycle vertices by g i : 1 ≤ i ≤ n .
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 Y n be the prism graph; then, the metric dimension of the line graph of the prism graph is 3 for n ≥ 3. Now, in the following theorem, we will compute the fault-tolerant metric dimension of the line graph of the prism graph.
Theorem 5. Let Y n be the prism graph; then, the fault-tolerant metric dimension of the line graph of the prism graph is 4 for n ≥ 3.
Proof. To prove this theorem, consider the following cases.

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Representation of vertices g i when n � 4 with respect to W ′ is Representation of vertices g i when n ≥ 6 with respect to W ′ is It can be easily seen that AD((s, t) | W ′ ) has at least two entries in the 4-vector different from zero for any s, t ∈ V(L(Y n )). Hence, W ′ is a fault-tolerant resolving set of L(Y n ). So, by using eorem 4 and Lemma 1, β ′ (L(Y n )) � 4 for every even n.