Abstract
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.
1. Introduction
The concept of metric dimension is applicable in all those networks where there is a need of localization of particular nodes. It is significantly used in different fields of science such as telecommunication, road networking, chemistry, and image processing to find winning combinations for different games. Slater in 1975 [1] introduced the locating set of graphs, whereas, in 1976, Harary and Melter [1] defined the term resolving set for graphs. Later on, both the terms were emerged and named as metric-based basis or generator. For a graph, is termed as metric generator if for , there exists a vertex , such that . Then, the vertex is said to distinguish (resolve) vertices a and b. If , then the distance coordinate vector of is s-tuple . This metric generator is extensively studied in literature.
The behavior of metric dimension of graphs relative to different graph products was investigated by different 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 different combinations of complete, path, cycle graphs by Sunitha et al. [4]. Even though metric generator was first proposed for problem of robot navigation, now this metric generator along with its various variants can have interesting and significant connections to other fields as well. For example, the determination of local variant named as local metric dimension may be associated to limited function of robot sensors. The mixed metric generator is one of the variants of the metric generator. This mixed version of metric generators was presented in 2017 by Kelenc et al. [5]. A subset is termed as the mixed metric generator of graph if coordinate distance vectors relative to of any two distinct elements of are not same. The smallest set which is the mixed generator of graph is termed as mixed basis, and its cardinality is termed as mixed dimension . Kelenc et al. [5] showed that a necessary and sufficient condition for a graph of order r to have mixed dimension two is . They also proved that and for .
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 sufficient to localize all stations or places uniquely and to deliver proper information to confront the problem. This 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.
The 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. They exhibited that for a graph of order m, where P is the fault-tolerant partition dimension and D is the diameter of the graph. The 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. The motivation of this research work is to fill this literature gap and to apply the concept of fault-tolerance in mixed generators.
2. Mixed Dimension of Rooted Product
The rooted product can be constructed by taking connected graphs and and a root vertex such that V(Q1 o Q2) = {(a1, b1)|a1 ∈ V(Q1), b1 ∈ V(Q2)} and E(Q1 o Q2) = {(a1, b1)(a2, b2)|a1 = a2 & b1 ∼ b2 or b1 = b2 = o & a1 ∼ a2}.
Corollary 1. (see [10]). A pendant vertex always belongs to every mixed metric generator of the graph.
Theorem 1. Consider a graph Q having order n ≥ 2. If o is the pendant vertex of , then .
Proof. Let and be the mixed metric basis of . Also, suppose that , such that and are the pendant vertices and . To construct required , take the vertex . The vertex set and edge set of are given as and . Clearly, the set of pendant vertices of is . If is the mixed metric generator of , then from Corollary 1, . Hence,Let . Then, . Now, we prove that M is a mixed metric generator. For this, let . Then, there arise three cases.
Case 1. Let . If and , such that ; then, If and , then . If and , then clearly, . But if and , then . Now, as n ≥ 2, there exist , such that . But as is nearer to as compared to , so . This shows thatThis implies that . Now, if and , then as is nearer to as compared to (pi, qk−1), so . This shows thatThis further shows that . Thus, every pair is resolved by M.
Case 2. Let . Since is the mixed resolving set of , there exists , such thatConsider the pendant vertex . Then, any path from (respectively h) to must contain andAs , so . Thus, all elements of are resolved by M.
Case 3. Last, let , . Then, or . In any case, but , which shows that the pair in this case are also resolved by M. Hence, M is a mixed metric generator and .
Now, we note that by considering o as the pendant vertex of P2, the corona graph is actually the rooted product graph of Q by P2, i.e., .
As a consequence, the following corollary can be stated.
Corollary 2. If is a corona graph with t vertices, then .
3. Mixed Fault-Tolerant Generators of Graphs
A mixed metric generator of a graph J is termed as the mixed fault-tolerant (MFTM) generator, if for , u is also a mixed generator. The smallest MFTM generator is termed as MFTM basis, and its cardinality is MFTM dimension .
Remark 1. From definition, 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 with labeling as shown in Figure 1 and . Then, the distance coordinate vectors of all vertices of relative to are computed as follows:Similarly, the distance coordinate vectors of all of its edges relative to are computed as follows:We can see that all distance coordinate vectors are distinguished by at least two points. Thus, is a MFTM generator, and . But from Proposition 4.4 of [10], . This along with [7] implies that . Thus, we have .
To recall, for the vertex in a graph , the open neighbourhood is the collection of all vertices adjacent to and closed neighbourhood is . For a vertex , a vertex is referred to as maximal neighbourhood of if . A vertex is referred to as the dominant vertex if it is adjacent to all other vertices of graph.
Lemma 1. (see [10]). Consider and . Suppose for each , there is , such that . Then, is a mixed metric generator for .
Lemma 2. (See [6]). For a graph , let be a metric generator and for . Then, is a FTM generator.
Theorem 2. A connected graph of order at least 3 contains a mixed fault-tolerant (MFTM) generator if and only every vertex is neither a pendant vertex nor possesses a maximal neighbourhood.
Proof. Let be a MFTM generator of connected graph . Suppose is a pendant vertex of . Then, there is a unique pendant edge with one end vertex as , say . Then, by Corollary 1, i.e., for some . Now, assume that the distance coordinate vector of relative to is given asAs is adjacent to and , therefore th coordinate of is equal to 1. Since is a pendant vertex, every path from any vertex of to must contain vertex . This implies that is nearer to any vertex of than . This further implies that for , we have but . Thus, the distance coordinate vector of edge relative to is given asThe th coordinate is 0 because and . It is cleared from [3, 11] that the coordinate vectors of and differ exactly by one coordinate. This further implies that . This shows that cannot be a MFTM generator, which leads to a contradiction. Next, we may assume that the graph has no pendant vertex, but there exists a vertex t having maximal neighbourhood c, i.e., . This implies that but . The elements and can only be distinguished by the vertex because if there is a vertex , such that and , then as , soThis further implies that any shortest path from to must have a vertex from different from . As , there is a vertex on the shortest path from to . Since , c ∼ . Therefore,This contradicts (6). Thus, and c can only be distinguished by t. Now, if , then cannot be a mixed generator. On the other hand, if , then cannot not be the mixed generator. In any case, is not a MFTM generator which again leads to a contradiction. Conversely, suppose that any vertex of is the nonpendant vertex and does not have any maximal neighbourhood, i.e., for and for any . For any , we exhibit that the set is a mixed generator. Let . As and (by assumption), there is a vertex (different from and ), such that but . Then, but . This shows that . Using Lemma 1, it is easy to see that is a mixed generator for any . Thus, the vertex set itself is the MFTM generator.
Corollary 3. A tree does not possess the mixed fault-tolerant generator.
Proof. Since a tree must have minimum two pendant vertices, the result follows using Theorem 2.
Corollary 4. A connected graph containing a dominant vertex d have no mixed fault-tolerant generator.
Proof. Let d be a dominant vertex of connected graph . Then, d serves as maximal neighbourhood of every vertex of . The result follows by using Theorem 2.
Theorem 3. Suppose is a graph having girth at least four and contains the MFTM generator. If is a mixed basis of graph J, then is a MFTM generator for .
Proof. Suppose . If , then as , so is a mixed generator. Thus, we may assume that . Now, we show that still is a mixed generator for . For this, take two elements . Now, there arise the following cases: Case (1): suppose , such that . As and both are vertices, so using Lemma 1, these can be distinguished by a vertex of , where and To show and are distinguished by some element of , it is enough to exhibit that . To show this, take Then, clearly, , that is for every . This implies that for every or we can say that . Case (2): now suppose and are the two distinct edges, such that and . Clearly, one of must be different from one of . Suppose there does not exist any vertex in that distinguished these edges. Then, as is mixed metric and , so the edges and must be distinguished by the vertex a, that is, d(a, ). Assume without any loss of generality (WLG) that 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 , and are distinct, i.e., edges are nonadjacent. Using (13), we have Assume WLG that and are nearer to the vertex relative to and , respectively. Then, from (14), we have Now assume that and lies on the minimum path between and . Then, clearly, ∈ Fm \ {a} being member of neighbourhood of , we can write By using (15) and (16), we have Now, by taking the path , we have . Then, using (17), we have Since and lies on the minimum path between and a and , therefore . Now, by taking the path , we have . This implies that . Thus, we can write Also, we have We claim that for otherwise, which contradicts [10]. Hence, , and using [1, 4, 13], we can write Which implies that edges and h are distinguished by . Now, if coincides with , then , such that and . Hence, and h are distinguished by . Case 2(b): now suppose that and h are adjacent and is their common vertex, i.e., and . Furthermore, suppose that a and are distinct. This implies that and . Then, (13) becomes If lies on the minimum path between a and , then we have Also, we have We claim that ; for otherwise, we have This contradicts (22). Thus, , and using (23) and (24), we can write Thus, the edges are distinguished by . Now, suppose a lies on one of the edges and h. Using (13), it is easy to see that only possibility is that the vertex a coincides with vertex . If the vertex , then and h are distinguished by . Thus, we may suppose that . This further implies that there exists a vertex , such that . If , then we have a triangle , which is not possible as the graph has girth at least four. Thus, which implies that , but . Hence, this case is completed. Case (3): finally, suppose that is a vertex and is an edge of , such that there does not exist any vertex in that distinguishes them. Then, and h can only be distinguished by , i.e., Now, the vertex may lie on h or not. Case 3a: suppose and is distinct from , , and . Furthermore, suppose If the vertex is nearer to as compared to , then from (27), we have Consider on the minimum path between a and . Then, , and from (28), we have From the path , we can see that Now, using (29) and (30), We claim that ; for otherwise, , that is, , which contradicts (27). Thus, Using (31), (32), and (33), we can write This implies that the vertex and edge h are distinguished by . The case can be dealt using same arguments. Now, suppose that a = . If or , then and h are distinguished by (or ). Therefore, we may suppose that and . Then, there exists , such that . By using the paths and , we can see that Since a = and , so , i.e., resolves and h. Now, assume that vertex a coincides with some vertex of h, say , i.e., . If , then and h are distinguished by . But if , then is not adjacent to some vertex . This implies that and . This completes this subcase. Case 3b: suppose that the vertex lies on h. Let . If , then we have Thus, If , then and h are distinguished by but if , then . This implies that there exists a vertex on the minimum path between and . Using (37), we have This implies that . Hence, and h are distinguished by . Finally, suppose that the vertices a and both are on the edge h, that is, . As the graph possesses the MFTM generator, so by Theorem 2, the vertex a is not a pendant vertex and therefore . This further implies that there exists a vertex other than , i.e., . As has girth at least four, so c is not adjacent to ,and we have , whereas . Thus, in all cases, any pair of element of J is distinguished by vertices of . This completes the proof.
4. Conclusion
In this study, it is shown that the mixed metric dimension for the rooted product of graph of order 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. The 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. This 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 , where and are the closed and open neighbourhoods, respectively, and A is the mixed basis for Q.
Data Availability
The data used to support the findings of this study are included within the article and are available from the corresponding author upon request.
Conflicts of Interest
The authors declare that they have no conflicts of interest.