On Fault-Tolerant Resolving Sets of Some Families of Ladder Networks

. In computer networks, vertices represent hosts or servers, and edges represent as the connecting medium between them. In localization, some special vertices (resolving sets) are selected to locate the position of all vertices in a computer network. If an arbitrary vertex stopped working and selected vertices still remain the resolving set, then the chosen set is called as the fault-tolerant resolving set. The least number of vertices in such resolving sets is called the fault-tolerant metric dimension of the network. Because of the variety of applications of the metric dimension in diﬀerent areas of sciences, many generalizations were proposed, and fault tolerant is one of them. In this paper, we computed the fault-tolerant metric dimension of triangular snake, ladder, Mobius ladder, and hexagonal ladder networks. It is important to observe that, in all these classes of networks, the fault-tolerant metric dimension and metric dimension diﬀer by 1.


Introduction
Let G � (V(G), E(G)) be a simple connected graph, where V(G) and E(G) are the set of vertices and edges, respectively. e distance d(u 1 , u 2 ) between a pair of vertices u 1 , u 2 ∈ V(G) is the length of the shortest path joining them. A vertex u resolves or distinguishes a pair u 1 , u 2 if d(u 1 , u) ≠ d(u 2 , u). e representation of an arbitrary vertex u corresponding to an ordered nonempty subset of vertices Q � q 1 , q 2 , . . . , q h ⊂ V(G) is the h-component vector r(u|Q) � (d(u, q 1 ), d(u, q 2 ), . . . , d(u, q h )). A subset Q ⊂ V(G) is named the resolving set if any pair u 1 , u 2 ∈ V(G) possesses the condition of unique representation, that is, r(u 1 |Q) ≠ r(u 2 |Q). e least possible cardinality of Q is named the metric dimension of the graph G and symbolized as dim(G). A resolving set containing a least possible number of elements is named as basis. A basis set Q f of G is called the fault-tolerant basis set if, for every u ∈ Q f , the subset (Q f /u) again resolves the vertices of G. e least number of vertices in Q f is termed as the faulttolerant metric dimension of G and is denoted by dim f (G). Chaudhry et al. [1] proved an important relationship between the fault-tolerant metric dimension and the metric dimension of a graph G. (1) First time, the idea of metric dimension was studied by Slater [2] in 1975 and later by Harary and Melter [3] in 1976. Chartrand et al. [4] considered metric bases as sensors. If any of the sensors did not operate correctly, then we do not have sufficient knowledge to deal with the intruders. Hernando et al. [5] presented the idea of faulttolerant metric dimension to overcome this kind of problems. Fault-tolerant basis set delivers accurate information despite being one of the sensors is not working.
Because of different applications, the study of fault-tolerant resolving sets of different networks is as sundry as the study of the metric dimension is.
It was proved that computing the metric dimension is NP-complete [6]. However, the idea of uniquely identifying every vertex in a graph based on distance is very useful. e application of this concept in chemical structures was presented by Chartrand el al. [4]. In robot navigation, the work by Khuller et al. [7] motivated the researchers for the theoretical investigation of the metric dimension. e resolving set has found a number of applications. For more details, see [8][9][10]. Generally, computing the fault-tolerant metric dimension is also an NP-complete problem. Computing the fault-tolerant metric dimension is considered as one of the interesting but difficult problems in combinatorics. So far, only few structures have been investigated. In the initial paper, Hernando et al. [5] computed the faulttolerant basis sets of tree T. For a graph G, they derived a connection between the fault-tolerant metric dimension and the metric dimension, that is, dim f (G) ≤ dim(G) (1 + 2 × 5 dim(G)− 1 ). Javaid et al. [11] studied this parameter of cycle graph C n and discussed some bounds on the partition dimension. For more results related to the fault-tolerant metric dimension, the interested readers can see [12][13][14][15][16][17][18][19][20][21][22][23].
e core objective of this paper is to find some classes of graphs satisfying the equation dim(G) � dim f (G) + 1. It has been observed that if G is ladder (L m , m ≥ 3), Möbius ladder (ML m , m ≥ 4), and hexagonal Möbius ladder graph (HML m , m ≥ 2), then the metric dimension and fault-tolerant metric dimension differ by 1. Also, for triangular snake graph TS n , the above equality holds if and only if n � 5, 7. In order to prove these results, we need some results from the literature.
Theorem 1 (see [4]). e metric dimension of a graph is 1 iff G � P n .

Fault-Tolerant Metric Dimension of the Triangular Snake Graph
is section deals with the metric dimension and faulttolerant metric dimension of the triangular snake graph. Let P n be a path with vertex set V(P n ) � v 1 , v 2 , . . . , v n , where n � 2s + 1 is an odd integer. A triangular snake graph TS n is constructed from P n by connecting v 2i−1 with v 2i+1 for 1 ≤ i ≤ s. Note that s represents the cardinality of triangles in TS n . e triangular snake graph is depicted in Figure 1. In the following Lemma 1, we calculate the metric dimension dim(TS n ) of triangular snake graph TS n . Lemma 1. Let TS n be a triangular snake graph with s ≥ 2. en, Proof. From eorem 1, it follows that dim(TS n ) ≥ 2. To prove that dim(TS n ) � 2, we prove that dim(TS n ) ≤ 2. Let Q � v 1 , v n be a resolving set. en, the representations of vertices v t regarding the resolving set Q are as follows: As all the vertices have unique representations regarding the resolving set Q, hence, dim(TS n ) ≤ 2. □ Theorem 5. Let TS n be a triangular snake graph; then, Proof. First, we show that dim f (TS n ) � 3 when s � 2, 3. To prove this, it is enough to show that dim f (TS n ) ≤ 3. e inequality dim f (TS n ) ≥ 3 follows from Lemma 1 and equation (1). If we take Q f � v 1 , v 4 , v 2s+1 , then it satisfies the condition of the fault-tolerant resolving set. e following is the representation of any vertex v t of the graph TS n with respect to Q f : where z � 1 when t � 3, 5 and zero, otherwise. Now, we prove that dim f (TS n ) ≤ 4 when s ≥ 4. For this, we construct a fault-tolerant resolving set. Take   2 Complexity ; the following is the representation of any vertex of graph with Q f : where z � 1 for t � even and zero, otherwise. Hence, it follows from the above discussion that dim f (TS n ) ≤ 4 since every vertex of TS n has a unique representation regarding resolving set Q f . We prove the reverse inequality dim f (TS n ) ≥ 4 by the contradiction method. Suppose dim f (TS n ) < 4, it follows from Lemma 1 that dim f (TS n ) � 3. Let us consider some resolving set Q f with cardinality 3.
In other words, we are taking the first vertex from the first triangle and the remaining vertices from any arbitrary triangle. en, it is easy to observe that there exists an integer 6 ≤ j ≤ n − 1 such that Hence, it follows from the above discussion that dim f (TS n ) ≠ 3 when s ≥ 4. is implies that dim f (TS n ) ≥ 4 when s ≥ 4. is completes the proof.

Fault-Tolerant Metric Dimension of Ladder Graphs
A ladder graph is obtained by the Cartesian product of path P n with path P 2 and is denoted by L m . Observe that m � 2n. e ladder graph is shown in Figure 2. In the following theorem, we compute the fault-tolerant metric dimension of the ladder graph. Proof.
e inequality dim f (L m ) ≥ 3 follows from eorem 4 and equation (1). To prove dim f (L m ) ≤ 3, we construct a fault-tolerant resolving set. Let Q f � v 1 , v 3 , v 2n−1 ; then, the representation of any vertex v t of the graph with Q f can be computed as where w � 1 for t � 2 and z � 1 for t � even and zero, otherwise. Observe that all the vertices of L m have a unique representation regarding fault-tolerant set Q f . Hence, dim f (L m ) � 3. e Möbius ladder was first introduced by Richard Guy and Frank Harary [26]. It is a three-regular graph created by a cycle C n by joining vertices u, v of the cycle iff d(u, v) � diam(C n ). Möbius ladder has an even number of vertices. It is denoted by M n , where n � 2m. e vertices which satisfy this condition are called antipodal. e graph of the Möbius ladder is shown in Figure 3. Möbius ladders have several uses in different fields of sciences such as stereochemistry, computer networks, and electrotechnology. In the next result, we find the fault-tolerant metric dimension of the Möbius ladder.
Proof. Let m ≥ 4 be an even integer. e inequality dim f (M n ) ≥ 4 follows from eorem 2 and equation (1). To prove dim f (M n ) ≤ 4, we construct a set satisfying the definition of fault-tolerant metric dimension. Let Q f � v 1 , v 2 , v m , v m+1 ; then, the distances of any vertex v t of the graph corresponding to Q f can be computed as follows.
Distance of vertex v t from vertex v 1 is Complexity Since all the vertices of M n have unique representations regarding the resolving set Q f , hence, dim f (M n ) � 4. Now, let m ≥ 5 be an odd integer. To prove that dim f (M n ) � 5, we construct a resolving set satisfying the condition of the fault-tolerant metric dimension. Let m+2 , v (3m+5/2) ; then, the distance of any vertices v t from the vertex v (m+5/2) , v m+2 , and v (3m+5/2) is as follows.
Distance of vertex v t from vertex v (m+5/2) is Since all the vertices of M n have unique representations regarding decided resolving set Q f , hence, dim f (M n ) � 5.
A hexagonal Möbius ladder HML(m, n) is constructed by subdividing each horizontal edge of the square lattice; thus, we obtained a lattice in which each cycle has order six; after creating this lattice, twist the recently attained lattice 180°and join the utmost left and right vertices as shown in Proof. Let m � 2; the inequality dim f (HML m ) ≥ 3 follows from eorem 3 and equation (1). To prove dim f (HML m ) ≤ 3, we construct a set satisfying the definition of fault-tolerant metric dimension. Let Q f � v 1 , v 4 , v 6 ; then, the distances of any vertex v t of the graph corresponding to Q f can be computed as follows.
Distance of vertex v t from vertex v 1 is Distance of vertex v t from vertex v 4 is Distance of vertex v t from vertex v 6 is It is easy to see that every vertex of HML m has its unique representation with the resolving set Q f . Hence, dim f (HML m ) � 2.
Let m ≥ 3; to prove that dim f (HML m ) � 4, it is enough to construct a fault-tolerant resolving set of four elements that satisfy the condition of fault-tolerant metric dimension. Let Q f � v 1 , v 2 , v 2m , v 2m+2 ; then, the distance of any vertex v t of HML m with Q f can be computed as follows.
Distance of vertex v t from vertex v 1 is Distance of vertex v t from vertex v 2m+2 is It is easy to see that all the vertices of HML m have unique representations regarding resolving set Q f . Hence, dim f (HML m ) � 3.

Conclusion
In this article, we study the fault-tolerant metric dimension of the triangular snake graph, ladder, Möbius ladder, and hexagonal Möbius ladder. It is known that, for cycle graph C n , we have dim f (C n ) � dim(C n ) + 1. erefore, it is interesting to find some classes of graph in which this equality holds. Note that, in case of the triangular snake graph, ladder, Möbius ladder, and hexagonal Möbius ladder, we have dim f � dim + 1.

Data Availability
No data were used to support this study.

Disclosure
is research was carried out as a part of the employment of the authors.

Conflicts of Interest
e authors declare no conflicts of interest.