[Retracted] Computing Edge Version of Resolvability and Double Resolvability of a Graph

Ahmad, Muhammad; Zahid, Zohaib; Rashid, Tabasam; Guirao, Juan Luis Garcia

doi:https://doi.org/10.1155/2022/2448032

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Abstract Introduction Preliminaries Conclusion Data Availability Conflicts of Interest Acknowledgments References Copyright Related Articles

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Volume 2022 | Article ID 2448032 | https://doi.org/10.1155/2022/2448032

[Retracted] Computing Edge Version of Resolvability and Double Resolvability of a Graph

Muhammad Ahmad,¹Zohaib Zahid,¹Tabasam Rashid,¹and Juan Luis Garcia Guirao^2,3,4

Academic Editor: Ajaya Kumar Singh

Received05 Oct 2021

Accepted17 Jan 2022

Published21 Feb 2022

Abstract

The field of graph theory is extensively used to investigate structure models in biology, computer programming, chemistry, and combinatorial optimization. In order to work with the chemical structure, chemists require a mathematical form of the compound. The chemical structure can be depicted using nodes (which represent the atom) and links (which represent the many types of bonds). As a result, a graph theoretic explanation of this problem is to give representations for the nodes of a graph such that different nodes have unique representations. This graph theoretic study is referred to as the metric dimension. In this article, we have computed the edge version of the metric dimension and doubly resolving sets for the family of cycle with chord for and .

1. Introduction

In research domains where networking is a basic and fundamental study block, graph theory is the most fundamental way to study and use these sciences. Consider the following scenario: (1) in network methods, database design, computer science, web document clustering, mobile phone networks, image processing, and resource management. (2) Only a few examples of biology include structures in cell biology, cell-sample sequencing, and bioinformatics. (3) In operations research, words such as minimal sum colouring, optimization, game theory, task, and time table scheduling are all employed. (4) Complex simulated structures of atoms, molecular bonds, chemoinformatics, and molecular descriptors are some of the study blocks in chemistry.

Graph invariants are excellent tools for analysing the abstract structures of graphs in a variety of disciplines. Nowadays, there are several distinct types of metric generators in graphs, each of which is explored both theoretically and practically, depending on its popularity or uses. One of them is the classical metric dimension which is used to detect devices in computer network difficulties.

For a connected graph , the distance between two vertices and is the minimum of the lengths of the paths of . A vertex is said to resolve two vertices and if . Suppose an ordered set of vertices = and be a vertex of , then the representation of with respect to is the -vector , also known as the distance vector. The set is called a resolving set for graph , if any vertex of has unique distance vectors with respect to . A metric basis for graph is a resolving set with a minimal number of entries. The number of entries in the metric basis is called its metric dimension (MD) and it is usually denoted by .

Combinatorics gave rise to the concept of the MD, and for the first time, Slater [1] proposed the problem of investigating the MD. Later, for the connected graph , Harary and Melter [2] suggested the same idea. The focus of the study was on a notion known as metric representation, which is a way of representing vertex locations in a graph. The resolving sets were initially created to find the precise position of an interrupter in a network, but later, Chartrand and Zhang [3] presented various applications in the fields of biology, chemistry, and robotics. The notion of trilateration may be applied to the MD of graphs from a two-dimensional real plane. For instance, three satellites in orbit use distances to calculate the precise location of an object on earth in the global positioning system (GPS). Several applications in chemical structure [4] and robot navigation [5] have been developed for the theoretical investigation of the MD. The MD of hamming graphs is linked to many coin weighing difficulties covered in [6, 7], as well as a detailed investigation of the game Mastermind given in [8]. This idea has been thoroughly investigated in a variety of domains, including combinatorial optimization [9], geographic routing protocols [10], and network discovery and verification [11].

Determining the precise value of the MD of arbitrary graphs is a computationally difficult problem. As a result, certain relevant bounds for various graph classes have been discovered. For example, Shao et al. evaluated the MD bounds for generalised Petersen graphs [12]. Chartrand et al. [4] categorised all graphs with MD , , and 1. Ahmad et al. [13] determined the minimal resolving sets for chorded cycles and kayak paddle graphs. The MD for various classes of convex polytopes is calculated by Imran and Baig [14, 15]. The MD of wheel related graphs (Jahangir graphs, respectively) were determined by Buczkowski et al. [16] (Tomescu and Javaid, respectively [17]). More details are discussed in [18, 19].

Finding the source of an epidemic in wide area networks can be a difficult task. For example, the only information utilized to identify a source of virus propagating throughout the network is the infection time of the subset of nodes termed as sensors. These sensors could be used to keep track of how long they have been infected. The only thing left is to figure out which sensors are needed to ensure the infection origin is precisely located. The solution to this difficulty is the property termed as double metric dimension [20, 21].

It may be simple to determine the mysterious virus source if the complete virus spreading process can be observed. Unfortunately, due to the high cost of data collection, the entire process may be too expensive. A doubly resolving sensor set may accurately identify infection sources even if the initial time of virus spread is unknown.

Caceres et al. [22] introduced the concept of doubly resolving sets (DRSs) in graphs during the investigation of the MD of the Cartesian product of graphs. For , are said to doubly resolve if . A subset is said to be a doubly resolving set (DRS) if some of its vertices doubly resolve every two vertices in . In other words, for every pair of vertices there exist for some and be a vector of length with all entries equal to 1. A doubly basis of graph is a DRS for with a minimum count, and its count is known as the double metric dimension (or doubly resolving number) of and is indicated by . Every DRS is obviously a resolving set; this implies .

Kratica et al. and Khuller et al. both demonstrated the NP-hardness of and in [5, 23], respectively. For the first time, Chen et al. [24] reported the explicit lower and upper bounds for the minimal DRS problem. Trees, cycles, and complete graphs were found to have a doubly resolving number by Caceres et al. [22]. Minimal DRSs for several families of Harary graphs have been established by Ahmad et al. [25]. MD and minimal DRSs were determined to be the same in various families of circulant graphs in [26]. Cangalovic et al. [27] and Kratica et al. [28] computed the doubly resolving number of prism graphs and hamming graphs, respectively. The doubly resolving number for several convex polytopes have been calculated in [29]. In [30], Pan et al. investigated the doubly resolving number for various types of convex polytopes. Liu et al. [31] studied the MD and minimal DRSs for the cocktail party graphs and jellyfish graphs. Further details are discussed in [32, 33].

Caceres et al. [22] found an upper limit in terms of and for the MD of the Cartesian product of graphs and . As a result, DRSs are essential when investigating Cartesian products of graphs. The idea of obtaining upper bounds of MD in the Cartesian product of graphs inspired us to work on DRSs of various graph families.

In this work, we investigated a newly defined variant of metric generators called the edge version of metric and double metric generators in order to contribute to the knowledge on these distance-related parameters in graphs. In classical metric dimension, the distance between vertices was of more importance, and in the newly defined variants, the distance between the edges has been given more weightage. Also, the edge version of doubly resolving sets (EVDRSs) is used to find the upper bound of the edge version of metric dimension (EVMD) in the same manner as in the classical metric dimension case. However, both variants are useful in many fields, for instance, to find the location of an intruder in a network, problems of pattern recognition, image processing, as well as the source of epidemics in social networks and in contagious diseases, etc.

2. Preliminaries

The line graph of an undirected graph is the graph whose vertices are in one-one correspondence with the edges of , and two vertices of the line graph are adjacent if and only if the corresponding edges in graph are adjacent. The edge distances can be calculated using the same way as the vertex distances. The EVMD is defined as follows [34].

Definition 1. (1)The edge distance between two edges and of is the minimum of the lengths of the paths in .(2)An edge is said to resolve two edges and of if .(3)Suppose an ordered set of edges and be an edge of , then the edge version of representation of with respect to is the -vector , also known as the edge distance vector.(4)The set is called an edge version of resolving set for graph , if any edge of has unique edge distance vector with respect to .(5)An edge version of the metric basis of graph is an edge version of the resolving set having a minimal number of edges. The represents the cardinal number of the edge version of the metric basis and is known as the edge version of the metric dimension of .

The EVDRSs presented in [35], is defined as follows.

Definition 2. (1)If , the edges and of the graph with size are supposed to edge doubly resolve edges and of the graph .(2)An ordered subset is called an edge version of doubly resolving set, if some of its edges edge doubly resolve every two edges in . In other words, for every pair of distinct edges , all coordinates of the vector cannot be same.(3)A doubly basis in the line graph is an edge version of doubly resolving set for with minimum count, and its count is termed as the edge version of double metric dimension of and indicated by .

It is important to note that the MD and DRSs of the line graph of a graph (particularly edges uniquely identifying edges) explored here are not the same as the parameter edge MD described in [36], which is vertices uniquely identifying edges.

Any EVDRS is also an edge version of resolving set, implying that for all graphs . The MD of line graphs has been extensively examined in mathematics [37, 38], and their importance in chemistry has been demonstrated [39–41]. Chartrand et al. studied the EVMD of path graphs in [4]. The EVMD was investigated in detail by Caceres et al. [22]. Eroh et al. used as bouquet and wheel graphs to determine in [42]. For the prism and -sunlet graphs, Ruby et al. [34] established the EVMD. Lv et al. determined the EVMD for the family of circulant graphs in [43]. The and of prism, n-sunlet, and kayak paddle graphs have been established by Ahmad et al. (further details are discussed in [35, 44]). Liu et al. explored the and of the necklace graph in [45]. Layer-sun graphs and their line graphs have also been explored in [46], for the minimal DRS problem. Further details are discussed in [47, 48].

The following is the structure of the rest of the article.

We discussed the graph in Section 3 and computed the for the chorded cycles . Section 4 calculates the for the chorded cycles . We conclude this paper at Section 5 by expressing an opinion.

3. The Edge Version of Metric Dimension for the Family of Cycle with Chord

A graph constructed from a cycle by combining two vertices whose distance in the cycle is , where , generates the family of cycle with chord (Figure 1).

We label the chord of by and the outer edges of by \{: \}, as demonstrated in Figure 1. The graph is also called the graph . Thus, it is sufficient to consider for given value of . The following result was recently computed for the MD of a cycle with chord .

Theorem 1 (see [13]). The metric dimension of the graphis 2, forand.

The following theorem computes the constant :

Theorem 2. The edge version of the metric dimension of the graphis 2, forand.

Proof. In order to calculate the EVMD, we prove the following four cases.

Case 1. Let be odd and be even.
Consider , and for each edge of , the edge version of representation with respect to is given as follows.
Here, is the edge version of the representation of with respect to , and the edge version of the representations of the edges are as follows:

Case 2. Let and be even.
Consider , and for each edge of , the edge version of representation with respect to is given as follows.
Here, is the edge version of the representation of with respect to , and the edge version of the representations of the edges are as follows:

Case 3. Let and be odd.
Consider , and for each edge of , the edge version of representation with respect to is given as follows.
Here, is the edge version of the representation of with respect to , and the edge version of the representations of the edges are as follows:

Case 4. Let be even and be odd.
Consider , and for each edge of , the edge version of representation with respect to is shown as follows.
Here, is the edge version of the representation of with respect to , and the edge version of the representations of the edges are as follows:We can see that the edge version of all the edges in the graph is unique, implying that . But the abovementioned graph is not a path graph implies that . As a result, we can conclude that is 2 in all four cases.

4. The Edge Version of Doubly Resolving Sets for the Family of Cycle with Chord

This section will focus to calculate the EVDRSs for the family of cycle with chord .

Theorem 1 implies that , for . We will also demonstrate that for .

To calculate the edge distances for the family of cycles with chord , the following steps are carried out.

Define be the edge set in at distance from . Table 1 can be easily formulated for , and it will be used to calculate the distances between any two edges in .

The symmetry of , where , shows that if and are even, we have the following:

If is odd and is even, we have the following:

If is even and is odd, we have the following:

If and are odd, we have the following:

As an outcome, if we know the distance for each , we can rebuild the distances between any two edges in .

Lemma 1. , for alland, whereand.

Proof. We know that for all and , . Therefore, we need to explain that every subset of cardinal number 2 is not an EVDRS for . Every set , as well as the nonedge doubly resolved pairs of edges from , are listed in Table 2. Let us explain that the edges are not edge doubly resolved by any two edges of the set . It is obvious that for , we have
, , and . So, , that is, the set is not an EVDRS of . All other forms of the set in Table 2 can be examined and confirmed to have nonedge doubly resolved pairs of edges as well.

Lemma 2. , for both and even, where and .

Proof. To prove , for both and even, it is sufficient to find an EVDRS having cardinal number 3. Now, from Table 1 using the sets , Table 3 illustrates the edge distance vectors for each vertex of in relation to the set .
From Table 3, we can verify that if two edges for some , then . Thus, if there are two edges and for any , and , then . Therefore, the set is the minimal EVDRS. Thus, Lemma 2 holds.

Lemma 3. , for oddand even, whereand.

Proof. To prove , for odd and even , it is sufficient to find an EVDRS having cardinal number 3. Now, from Table 1 using the sets , Table 4 illustrates the edge distance vectors for each vertex of in relation to the set .
From Table 4, we can verify that if two edges for some , then . Thus, if there are two edges and for any , and , then . Therefore, the set is the minimal EVDRS. Thus, Lemma 3 holds.

Lemma 4. , for evenand odd, whereand.

Proof. For , the minimal EVDRS is . To prove , for even and odd , it is sufficient to find an EVDRS having cardinal number 3. Now, from Table 1 using the sets , Table 5 illustrates the edge distance vectors for each vertex of (where ) in terms of the set .
From Table 5, we can verify that if two edges for some , then . Thus, if there are two edges and for any , and , then . Therefore, the set is the minimal EVDRS. Thus, Lemma 4 holds.

Lemma 5. , when bothandodd, whereand.

Proof. For , the minimal EVDRS is . In order to prove , for both and odd, it is sufficient to find an EVDRS of cardinality 3. Now, from Table 1 using the sets , Table 6 illustrates the edge distance vectors for each vertex of (where ) in terms of the set .
From Table 6, we can verify that if two edges for some , then . Thus, if there are two edges and for any , and , then . Therefore, the set is the minimal EVDRS. Thus, Lemma 5 holds.

It is displayed from the whole technique that for and . The set is the minimal EVDRS for and . Using Lemma 2–Lemma 5, the main theorem is stated as follows:

Theorem 3. Letbe a family of cycle with chord. Then, , for.

5. Conclusion

We computed the EVMD in this article by considering as a family of cycles with the chord . The edge version of the double metric dimension was also computed. The graph is interesting to study in this situation because the cardinality of minimal EVDRSs and its EVMD are not dependent on the number of vertices of these graphs.

Here we end with the following open problem: Open problem 5.1. Find the classes of graphs such that

Data Availability

The required data of this study are included within the article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

This research work was partially supported by Ministerio de Ciencia, Innovacin y Universidades grant no. PGC2018-097-198-B-I00 and Fundación Séneca. de la Region de Murcia grant no. 20 783/PI/18.

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