Computing Fault-Tolerant Metric Dimension of Connected Graphs

For a connected graph, the concept of metric dimension contributes an important role in computer networking and in the formation of chemical structures. Among the various types of the metric dimensions, the fault-tolerant metric dimension has attained much more attention by the researchers in the last decade. In this study, the mixed fault-tolerant dimension of rooted product of a graph with path graph with reference to a pendant vertex of path graph is determined. In general, the necessary and sufficient conditions for graphs of order at least 3 having mixed fault-tolerant generators are established. Moreover, the mixed fault-tolerant metric generator is determined for graphs having shortest cycle length at least 4.


Introduction
e concept of metric dimension is applicable in all those networks where there is a need of localization of particular nodes. It is signi cantly used in di erent elds of science such as telecommunication, road networking, chemistry, and image processing to nd winning combinations for di erent games. Slater in 1975 [1] introduced the locating set of graphs, whereas, in 1976, Harary and Melter [1] de ned the term resolving set for graphs. Later on, both the terms were emerged and named as metric-based basis or generator. For a graph, J, W⊆V(J) is termed as metric generator if for a, b ∈ V(J)(a ≠ b), there exists a vertex w ∈ W, such that d(a, w) ≠ d (b, w). en, the vertex w ∈ W is said to distinguish (resolve) vertices a and b. If W w 1 , . . . , w s , then the distance coordinate vector of a ∈ V(J) is s-tuple r(a|W) (d(a, w 1 ), . . . , d(a, w s )). is metric generator is extensively studied in literature. e behavior of metric dimension of graphs relative to di erent graph products was investigated by di erent authors, like Cartesian product by Caceres et al. [2], the join product as well as Cartesian product by Hernando et al. [3], and join of di erent combinations of complete, path, cycle graphs by Sunitha et al. [4]. Even though metric generator was rst proposed for problem of robot navigation, now this metric generator along with its various variants can have interesting and signi cant connections to other elds as well. For example, the determination of local variant named as local metric dimension may be associated to limited function of robot sensors. e mixed metric generator is one of the variants of the metric generator.
is mixed version of metric generators was presented in 2017 by Kelenc et al. [5]. A subset M⊆V(J) is termed as the mixed metric generator of graph J if coordinate distance vectors relative to M of any two distinct elements of V(J) ∪ E(J) are not same. e smallest set which is the mixed generator of graph J is termed as mixed basis, and its cardinality is termed as mixed dimension (dim(J)). Kelenc et al. [5] showed that a necessary and su cient condition for a graph J of order r to have mixed dimension two is J P r . ey also proved that dim(C n ) 3 and dim(P r 2 □P r 2 ) 3 for r 1 ≥ r 2 ≥ 2.
Hernando et al. [6] in 2008 presented the idea of the fault-tolerant metric (FTM) generator. To illustrate this, consider a network where the metric basis represents the censors [7]. In this situation, if some censor is forced to not operate properly, then other censors will not be su cient to localize all stations or places uniquely and to deliver proper information to confront the problem. is type of frequently occurring situation in networks was resolved by Hernando et al. [6] with the help of applying fault-tolerance in the metric generator. To consider the fault-tolerance in the metric generator, the new generator will be able to transfer the information correctly even when a censor is disabled for any reason. It can be said that this FTM generator is applicable to all those networks where metric dimension has its significance like in optimal flow control problems of interconnecting networks. e FTM generator and FTM dimension for a tree was determined in [6]. Javaid et al. [8] also explored the FTM dimension and FTM partition dimension of different graphs. ey exhibited that for a graph of order m, m ≤ PD n− 2 m, where P is the fault-tolerant partition dimension and D is the diameter of the graph. e FTM dimension of infinite families of convex polytopes is proved to be constant by Raza et al. [9]. So far, a lot of research works on metric dimension and its different variants have been carried out by different authors, whereas comparatively less investigations have been made in the exploration of mixed generators. e motivation of this research work is to fill this literature gap and to apply the concept of fault-tolerance in mixed generators.

Mixed Dimension of Rooted Product
e rooted product Q 1 ⊳ o Q 2 can be constructed by taking connected graphs Q 1 and Q 2 and a root vertex o ∈ Q 2 such that Corollary 1 (see [10] Proof. Let V(Q) � p 1 , p 2 , . . . , p n and S � w 1 , w 2 , . . . , w k be the mixed metric basis of Q. Also, suppose that V(P m ) � q 0 , q 1 , . . . , q m− 1 , such that q 0 and q m− 1 are the pendant vertices and and Let X � (p i , q j ), (p i , q j− 1 )(p i , q j ): 1 ≤ i ≤ n, 1 ≤ j ≤ m − 1}⊆Q ′ . en, Q � Q ′ − X. Now, we prove that M is a mixed metric generator. For this, let g, h ∈ Q ′ . en, there arise three cases. □ Case 1: Let g, h ∈ X. If g � (p i , q k )(respectively (p i , q k − 1 ) (p i , q k )) and h � (p j , q k )(respectively (p j , q l− 1 )(p j , q l )), Consider the pendant vertex (w i , q m− 1 ) ∈ S * . en, any path from g (respectively h) to (w i , q m− 1 ) must contain s � (w i , o) and ). us, all elements of Q are resolved by M.
which shows that the pair g, h in this case are also resolved by M. Hence, M is a mixed metric generator and dim m (Q ′ ) � n. Now, we note that by considering o as the pendant vertex of P 2 , the corona graph Q ′ � QoK 1 is actually the rooted product graph of Q by P2, i.e., QoK 1 � Q⊳ O P 2 .
As a consequence, the following corollary can be stated.

Mixed Fault-Tolerant Generators of Graphs
is also a mixed generator. e smallest MFTM generator is termed as MFTM basis, and its cardinality is MFTM dimension (dim m f (J)). Remark 1. From de nition, it is obvious that the MFTM dimension is always greater than or equal to the mixed metric dimension, i.e., for any graph J, Example 1. Take P 3 □P 3 with labeling as shown in Figure 1 and . en, the distance coordinate vectors of all vertices of P 3 □P 3 relative to M are computed as follows: r v 9 |M (4, 2, 2, 0).

(7)
Similarly, the distance coordinate vectors of all of its edges relative to M are computed as follows:

(8)
We can see that all distance coordinate vectors are distinguished by at least two points. us, M is a MFTM generator, and dim m f (P 3 □P 3 ) ≤ 4. But from Proposition 4.4 of [10], dim m (P 3 □P 3 ) 3. is along with [7] implies that To en, there is a unique pendant edge with one end vertex as p 1 , say p p 1 p 2 . en, p 2 ∈ M by Corollary 1, i.e., p 2 q j for some j 1, . . . , s. Now, assume that the distance coordinate vector of p 1 relative to M is given as As p 1 is adjacent to p 2 and p 2 q j , therefore j th coordinate of r(p 1 |M) is equal to 1. Since p 2 is a pendant vertex, every path from any vertex of J to p 2 must contain vertex p 1 . is implies that p 1 is nearer to any vertex of J than p 2 . is further implies that for (p 2 , p). us, the distance coordinate vector of edge p relative to M is given as e j th coordinate is 0 because p p 1 p 2 and p 2 q j . It is cleared from [3,11] that the coordinate vectors of p 1 and p di er exactly by one coordinate. is further implies that r(p 1 |M∖ p 2 ) r(p|M∖ p 2 ). is shows that M cannot be a MFTM generator, which leads to a contradiction. Next, we may assume that the graph J has no pendant vertex, but there exists a vertex t having maximal neighbourhood c, i.e., d(c, t) 1 but d(tc, t) 0. e elements tc and c can only be distinguished by the vertex t because if there is a vertex m ∈ M, such that m ≠ t and

Journal of Mathematics d(c, m) ≠ d(m, tc), then as d(m, tc) � min(d(t, m), d(c, m)), so d(t, m) < d(c, m). (11)
is further implies that any shortest path from m to t must have a vertex from N(t) different from c.

Using Lemma 1, it is easy to see that V(J)\ a
{ } is a mixed generator for any a ∈ V(J). us, the vertex set V(J) itself is the MFTM generator. Proof. Since a tree must have minimum two pendant vertices, the result follows using eorem 2.   (2): now suppose g and h are the two distinct edges, such that g � g 1 g 2 and h � h 1 h 2 . Clearly, one of g i must be different from one of h j . Suppose there does not exist any vertex in M f \ a { } that distinguished these edges. en, as M is mixed metric and M⊆M f , so the edges g and h must be distinguished by the vertex a, that is, d(a, g) ≠ d(a, h) d(a, g). Assume without any loss of generality (WLG) that d(a, g) < d(a, h). (13) Now, there are two possibilities, either all vertices on these edges are distinct or there is some common vertex between them.
Case 2(a): suppose g 1 , g 2 , h 1 , and h 2 are distinct, i.e., edges are nonadjacent. Using (13), we have Assume WLG that g 1 and h 2 are nearer to the vertex a relative to g 2 and h 1 , respectively. en, from (14), we have Now assume that g 1 ≠ a and a ′ ∈ N(a) lies on the minimum path between a and g 1 . en, clearly, a ′ ∈ M f \ a { } ∈ F m \ {a} being member of neighbourhood of a, we can write d a, g 1 � 1 + d a ′ , g 1 (16) By using (15) and (16), we have Now, by taking the path a − a ′ − · · · − h 2 , we have d(a, h 2 ). en, using (17), we have Since a ′ ∈ N(a) and lies on the minimum path between g 1 and a and d(a, g 1 ) < d(a, g 2 ), therefore d(a ′ , g 1 ) < d(a, g 1 ) < d(a, g 2 ). Now, by taking the path a − a ′ − · · · − g 2 , we have d(a ′ , g 1 ) < d(a, g 2 ) ≤ d(a ′ , g 2 ) + 1.

(19)
Also, we have (20) We claim that d(a ′ , g 1 ) < d(a ′ , h 1 ), for otherwise, [10]. Hence, d(a ′ , g 1 ) < d(a ′ , h 2 ), and using [1,4,13], we can write Which implies that edges g and h are distinguished by a ′ ∈ M f \ a { }. Now, if a coincides with g 1 , then g 2 ∈ M f \ a { }, such that d(g, g 2 ) � 0 and d(h, g 2 ) ≥ 1. Hence, g and h are distinguished by g 2 . Case 2(b): now suppose that g and h are adjacent and g 2 is their common vertex, i.e., g � g 1 g 2 and h � g 2 h 1 . Furthermore, suppose that a and g 1 are distinct. is implies that a ≠ g 1 , g 2 , h 1 and d(g 1 , a) < d(g 2 , a).
If a ′ ∈ N(a) lies on the minimum path between a and g 1 , then we have Also, we have We claim that d(a ′ , g 1 ) < d(a ′ , h 1 ); for otherwise, we have is contradicts (22). us, d(a ′ , g 1 ) < d(a ′ , h 1 ), and using (23) is further implies that there exists a vertex a ′ ∈ N(a), such that h 1 ≁a ′ . If g 2 ∼ a ′ , then we have a triangle g 1 − a ′ − g 2 − g 1 , which is not possible as the graph J has girth at least four. us, g 2 ≁a ′ which implies that d(a ′ , h) ≥ 2, but d(a ′ , g) � 1. Hence, this case is completed.
Case (3) Consider a ′ ∈ N(a) on the minimum path between a and g. en, d(g, a ′ ) � d(g, a) − 1, and from (28), we have From the path a − a ′ − · · · − h 1 , we can see that Now, using (29) and (30), ′ . By using the paths a − a ' ′ − · · · − h 2 and a − a ' ′ − · · · − h 1 , we can see that Since a � g and a ″ ∼ a, so d(g, a ′ )  ′ ∈ N(a) on the minimum path between a and h 2 . Using (37), we have is implies that d(b ′ , g) > d(b ′ , h 2 ). Hence, g and h are distinguished by b ' ′ ∈ M f \ a { }. Finally, suppose that the vertices a and g both are on the edge h, that is, h � ag. As the graph J possesses the MFTM generator, so by eorem 2, the vertex a is not a pendant vertex and therefore deg(a) > 1. is further implies that there exists a vertex c ∈ N(a)⊆M f \ a { } other than g, i.e., c ≠ g. As J has girth at least four, so c is not adjacent to g ,and we have d(c, h) � 1, whereas d(c, g) ≥ 2.
us, in all cases, any pair of element of J is distinguished by vertices of M f \ a { }. is completes the proof.

Conclusion
In this study, it is shown that the mixed metric dimension for the rooted product of graph of order n ≥ 2 n by path graph by taking pendant vertex as root vertex is n. As a consequence, it is presented that the mixed metric dimension of the corona graph with t vertices is t/2. e notion of the mixed fault-tolerant metric generator is defined for the mixed generator. As the graphs like path graph, tree, and complete graph do not have any mixed metric generators, so it is important to classify those graphs which possess the mixed generator. is problem is settled in this study, and the graphs having the mixed fault-tolerant metric generator are characterized. Specifically, it is shown that the necessary and sufficient conditions for existence of the mixed generator for a graph Q are that the graph Q does not have pendant vertices and does not contain any vertex having maximal neighbourhood. Moreover, the mixed fault-tolerant metric resolving set for a graph Q with girth at least 4 is presented as and N(t) are the closed and open neighbourhoods, respectively, and A is the mixed basis for Q.

Data Availability
e data used to support the findings of this study are included within the article and are available from the corresponding author upon request.