Fault-Tolerant Metric Dimension of Generalized Wheels and Convex Polytopes

For a graph G , an ordered set S ⊆ V ( G ) is called the resolving set of G , if the vector of distances to the vertices in S is distinct for every v ∈ V ( G ) . The minimum cardinality of S is termed as the metric dimension of G . S is called a fault-tolerant resolving set (FTRS) for G , if S \ v { } is still the resolving set ∀ v ∈ V ( G ) . The minimum cardinality of such a set is the fault-tolerant metric dimension (FTMD) of G . Due to enormous application in science such as mathematics and computer, the notion of the resolving set is being widely studied. In the present article, we focus on determining the FTMD of a generalized wheel graph. Moreover, a formula is developed for FTMD of a wheel and generalized wheels. Recently, some bounds of the FTMD of some of the convex polytopes have been computed, but here we come up with the exact values of the FTMD of two families of convex polytopes denoted as D k for k ≥ 4 and Q k for k ≥ 6. We prove that these families of convex polytopes have constant FTMD. This brings us to pose a natural open problem about the existence of a polytope having nonconstant FTMD.


Introduction
Computer networks and other similar systems can be given design of graphs in which vertices are the representor of nodes and edges as a medium of connection between them. e location of fault in the computer network is the vertex of a graph. is idea to identify each vertex in the network was initially given by Slater in [1] and then by Harary and Melter in [2] to control the fault in a better way. erefore, the concepts of locating numbers and locating sets were developed. In fact, the actual terminology appeared way back by Blumenthal [3] for standard Euclidean spaces. However, not many researchers were forced to think about these ideas because of linear continuum nature of these spaces. e resolving set and ideas, pretty similar to it, are contributing in diverse areas today merely because of their usage in discrete nature. ese ideas are utilized to recognize nodes in a network, connected joins in graphs, mastermind games [4], strategies for the network discovery, combinatorial optimization, applications of pattern recognition, pharmaceutical chemistry, image processing, and game theory. A collection of points can be identified by finding the distance from all nodes in a network or graph. us, the problem of robot navigation may be solved by finding a minimal set of fixed vertices. Such minimal sufficiently large set is a resolving set of the graph. e corresponding cardinality is termed as metric dimension.
For a connected, simple graph G, metric d G : { } is the distance between x and y and is denoted by d G (x, y). Let W � w 1 , w 2 , · · · , w k ⊆ V(G) be an ordered set, and let v ∈ V(G). en, r(v|W) is the representation of v with respect to W and is the k-tuple (d(v, w 1 ), d(v, w 2 ), · · · , d(v, w k )). If r(v|W) are distinct for distinct vertices of G, then we say W is a resolving set for G, see [5]. A resolving set with minimum cardinality is a basis of G, and its cardinality is termed as the metric dimension of G, denoted by β(G).
Buczkowski et al. computed the metric dimension of wheel W n and proved that ⌊2n + 2/5⌋ for n ≥ 7 [5]. In [6], Hernando et al. proved that for fan, the metric dimension is ⌊2n + 2/5⌋ for n ≥ 7, and for Jahangir graphs J 2n , Tomescu and Javaid proved that ⌊2n/3⌋ for all n ≥ 4, see [7]. In [8,9], the authors proved some results about the metric dimension of regular graphs. In [10], the metric dimension for a flower graph is computed by the authors. In [11], Chartrand et al. proved that a metric dimension is 1 iff graph is a path. In 2014, the authors in [12] computed the metric dimension for the generalized wheel graph. Ali et al. [13], in 2012, and Munir et al. [14], in 2017, studied about the metric dimension of Möbius ladders. For unicyclic graphs, Poisson and Zhang found metric dimension in [15]. If for a family (G n ) n≥1 of connected graphs, where exists a constant C > 0 such that β(G n ) ≤ C for all n ≥ 1, then the family of graphs is said to have bounded metric dimension; otherwise, it has unbounded metric dimension. If all graphs of a family have the same metric dimension (means do not depend on n), then that family is called a family with constant metric dimension [16]. For further study, one can see [17][18][19][20][21][22][23][24][25][26].
Recent advancement in the concept of metric dimension has paved a path for a new related idea known as fault tolerance. Suppose a network consists of n processing units. To form a fault-tolerant self-stable system, it is essential that if any single unit becomes fail, another chain of units (excluding the faulty unit) can replace the originally used chain. So, the design of this fault-tolerance empowers the system to continue working, possibly at a reduced level and opposed to flop totally.
{ } is also a resolving set, ∀v ∈ S � , and consequently, the minimum cardinality of such S � is termed as FTMD, denoted by β ′ (G). If the order of S � is equal to β ′ (G), then it is called a fault-tolerant metric basis. FTMD is constant for a family G of connected graphs if it is not dependent on any choice of the member of that family. Fault-tolerant designs are successfully used in computer sciences and engineering [27].
e study of fault-tolerant sets was inaugurated by Slater in [28]. Hernando et al. developed the concept of an FTMD in [29]. ey studied the FTMD of trees. ey also proved a very important result in the form of upper bound irrespective of the choice of graph which is given as β ′ (G) ≤ β(G)(1 + 2.5 β(G)−1 ). Javaid et. al. in [30,31] studied fault-tolerance in resolvability, and also, they computed FTMD of few classes of graphs. It is not difficult to conclude that β ′ (G) ≥ β(G) + 1. Few more results were found by Shabbir and Zamfirescu in [32] about triangular lattice networks.
In this article, we considered families of convex polytopes and studied the FTMD for them. A convex polytope is the convex-hull of an finite set in any Euclidean space. ese are significant graphs and hence have been under discussion from many years. For instance, Baca in [33,34] computed labellings and magic of convex polytopes. Nazeer et al. computed the formulae for the average eccentricity, eccentricity, center, and periphery for the convex polytopes in [35]. In [21], Imran et al. found that the metric dimension is constant for convex polytopes S k , T k , Q k , and D k . In this article, we take a step further and prove that D k and Q k have constant FTMD. We also discuss the FTMD of multilevel wheel and classical wheel graphs.

Main Results
In this section, we put together our main results. At first, we prove that FTMD of D k is 4. en, we move towards the case of Q k . In both cases, we use double inequality of sets. In the end, we compute the FTMD of multilevel wheel.

Fault-Tolerant Metric Dimension of the Graph of D k and
e graph A k is obtained using two regular k-gones say P and P � with the vertices u 1 , u 2 , · · · , u k and z 1 , z 2 , · · · , z k , respectively, in anticlockwise. We join any vertex u p of P with the vertices z p and z p+1 of P � by edges for p � 1, 2, · · · , k. e graph of the convex polytope D k is formed [33], by the dual graph of A k consisting of the vertices u 1 , u 2 , u 3 , · · · , u k , x 1 , x 2 , x 3 , · · · , x k ; y 1 , y 2 , y 3 , · · · , y k ; and z 1 , z 2 , z 3 , · · · , z k . It consists of 5sided and k-sided faces. For compete details, see Figure 1. e convex polytope Q k defined in [34] has faces of 3 sides, 4 sides, 5 sides, and n-sides. e vertex set for the graph of the convex polytope Q k is the same as that of the convex polytope D k , and also, Let us call the vertices u p , 1 ≤ p ≤ k, x p , 1 ≤ p ≤ k, y p , 1 ≤ p ≤ k, and z p , 1 ≤ p ≤ k as the inner vertices, the middle vertices, the interior vertices, and the outer vertices, respectively. e cycles induced by u p , 1 ≤ p ≤ k , x p , 1 ≤ p ≤ k , and z p , 1 ≤ p ≤ k are the inner cycle, the middle cycle, and the outer cycle respectively, as shown in Figure 2.

Theorem 1. For both type of convex polytopes
Case I: k is even: let k � 2n; n ∈ Z + , and n ≥ 3. Let F � u 1 , u 2 , un + 1, u n+2 . We shall prove that F is a fault-tolerant metric basis for D k and Q k in the present case. We prove that the metric representations of all vertices of D k and Q k with respect to F are distinct in at least two coordinates. We call u k+1 , x k+1 , y k+1 , and z k+1 as the vertices of inner cycle, the inner vertices, the outer vertices, and the vertices of outer cycle, respectively. e metric representations of vertices lying on the inner cycle are e metric representations of vertices belonging to inner vertices are if n � 1, e metric representations of vertices belonging to outer vertices are r y p |F � (2, 2, n + 1, n + 1), if p � 1, e metric representations of vertices lying on the outer cycle are if p � 1, ese representations are distinct in at least two coordinates. So, F is an FTRS. erefore, for both (D k ) and (Q k ), β ′ ≤ 4. Since for both (D k ) and (Q k ), β � 3 as proved in [21], so β ′ > 3. Hence, β ′ � 4 when k is even. Case II: k is an odd integer: let k � 2n + 1 where n ∈ Z + and n ≥ 3, and F � u 1 , u 2 , u n+1 , u n+2 . We shall prove that F is a fault-tolerant metric basis for D k and Q k in the present case. We prove that the metric representations of all vertices of D k and Q k with respect to F are distinct in at least two coordinates. e metric representations of vertices lying on the inner cycle are e metric representations of inner vertices are r x p |F � (1, 2, n + 1, n + 1), if p � 1, e metric representations of outer vertices are r y p |F � (2, 2, n + 1, n + 2), if p � 1, e metric representations of vertices on the outer cycle are r z p |F � (3, 3, 2 + n, n + 3), if p � 1, ese representations are distinct in at least two coordinates. So, F is an FTRS. erefore, for both (D k ) and (Q k ), β ′ ≤ 4. Since for both (D k ) and (Q k ), β � 3 as proved in [21], so β ′ > 3. Hence, β ′ � 4 when k is odd.

Fault Tolerance in Metric Dimension of Multilevel Wheel.
In this part, we focus on the FTMD of multilevel wheel (W n,m ). An m wheel graph of size n is defined as follows: an m wheel graph of size n can be composed of mC n + K 1 . It consists of m cycles of size n, where the vertices of the m cycles are all connected to a common hub. us, W n,m has nm + 1 vertices, the center, and n rim vertices and has diameter 2. Figure 3 represents the graph of W 8,m .
A classical wheel graph is simply a specialized wheel with a single level denoted as W n,1 or simply W n . Here, we give the graph of W 8,1 in Figure 4.
We do not include the central vertex in the representation of the above graph W 8,1 . In the above graph, the vertices of the Since the vertices of G have distinct representations with respect to R 1 , R 1 is a resolving set. Since metric dimension of a wheel graph W 8 is 3, R 1 is a basis for G because it has metric dimension 3. Similarly, R 2 , R 3 , and R 4 are also basis for G with metric dimension 3. Now, We see that R 5 \ v { } is also a resolving set for each v in R 5 . As β(W 8,1 ) of the above example is 3, therefore, β , (W 8,1 ) � 4. Hence, R 5 is a fault-tolerant metric basis with β , (W 8,1 ) � 4, so R 5 is an FTMS.
If B � is a resolving set of the graph G, which contains two or more vertices of G, and we suppose that the vertices of B � � v j 1 , v j 2 , · · · , v j r , so the j 1 < j 2 < · · · < j r . We shall say that the vertices v j a , v j a + 1 for i � 1, 2, · · · , r − 1, and v j r , v j 1 are neighboring vertices. Now, the vertices lying between these neighboring vertices are referred to as the gap of B � and the number of vertices in a gap will be referred to as the size of the gap. If |B � | is t, then it has t gapes, and some of which may be empty. For example, the gap between v 2 and v 4 in Figure 4 is 1 because there is only one vertex v 3 in between v 2 and v 4 . Now, let B is an FTMS of (W n,m ). If we do not include the central vertex, then we make the following two conditions: (1) If n is even, then every gap of B contains exactly one vertex. Our aim in this section is to compute the FTMD of wheel W n and generalize the wheel graph W n,r . In [5], it was shown that dim(W n ) � ⌈(2n + 5)/2⌉, for every n ≥ 7. Now, suppose that B ′ is any fault-tolerant metric basis of W n . We observe the following results in the form of lemmas and theorems. Figure 4: A graph of W (8,1) wheel.

Lemma 1. If n is even and n ≥ 8, then every gap of B ′ (faulttolerant basis) contains exactly one vertex.
Proof.
(i) Suppose on the contrary, there exists a gap which contains two vertices and at least one of its neighboring gaps contains one vertex. en,   Proof. Suppose on the contrary, there exists a gap which contains two vertices. en, Proof. We see from Lemmas 1 and 2 that B ′ can be an FTMS because for each v in B ′ , the set B ′ \ v { } is a resolving set. Now, we see that the set B ′ \ v 1 , v 3 or B ′ \ v 2 , v 4 is not a resolving set, which is a contradiction. Hence, B ′ is a fault-tolerant metric basis. □ Lemma 4. If C 1 and C 2 denote the vertices of circle 1 and circle 2, respectively, then fault-tolerant metric basis for C 1 and C 2 is the same.
Proof. Let C 1 and C 2 denote the set of vertices of circle 1 and circle 2, respectively. Let v i ∈ C 1 and w i ∈ C 2 . Now, Also, there does not exist any v i belongs to C 1 such that by B ′ \ v i , we get r(v j /B ′ ) � r(v r /B ′ ) � (2, 2, 2, · · ·, 2). Similarly, there does not exist any w i belongs to C 2 such that by B ′ \ v i , we get r(w j /B ′ ) � r(w r /B ′ ) � (2, 2, 2, · · ·, 2). In the same way, we can show that all levels have the same FTMS. Now, we move towards the main results of this section. We divide the proof into the following two cases based on parity of n.
We shall show that it is true for m � k + 1. We have β ′ (W n,k+1 ) � β ′ (W n,k ) + β ′ (W n,1 ). en, Hence, the given statement is true for all positive integers m ≥ 3. We see from eorem 3 that the FTMD of W (11,1) is equal to 6 and the fault-tolerant metric basis for EW (11,1) is 10 . We see that B � v 1 , v 4 , v 6 , v 10 is a metric basis for W (11,1) with cardinality 4. In order to find fault-tolerant metric basis with cardinality 5, we need other four metric bases having cardinality 4 other than 10 . However, W (11,1) has only one metric basis with cardinality 4. It means that there is no faulttolerant metric basis with cardinality 5. So, we need to add two vertices to find fault-tolerant metric basis. Hence, B � v 1 , v 2 , v 4 , v 6 , v 8 , v 10 is a fault-tolerant metric basis with cardinality 6, and it also satisfies Lemma 1.

Example 2.
We also give another example to compute the FTMD of the EW (11,2) wheel graph and verify.

Conclusions and Open Problem
In this article, we computed the FTMD of graphs of two convex polytopes denoted by D n and Q n . We conclude that all these graphs have constant FTMD.

Theorem 3.
e fault-tolerant metric dimension of convex polytopes D n is 4 for n ≥ 4.

Theorem 4.
e fault-tolerant metric dimension of convex polytopes Q n is 4 for n ≥ 6.
We pose a natural open problem: Does there exist the graph of a convex polytope which has nonconstant FTMD? We also computed the FTMD of generalized wheel and arrive in the following theorem.
In Table 1, we presented the comparison between the metric dimension and fault-tolerant metric dimension for generalized wheel graphs.

Data Availability
No data were used to support the findings of this study.

Conflicts of Interest
e authors declare no conflicts of interest.