[Retracted] On Adjacency Metric Dimension of Some Families of Graph

Koam, Ali N. A.; Ahmad, Ali; Azeem, Muhammad; Khalil, Adnan; Nadeem, Muhammad Faisal

doi:https://doi.org/10.1155/2022/6906316

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Research Article | Open Access

Volume 2022 | Article ID 6906316 | https://doi.org/10.1155/2022/6906316

[Retracted] On Adjacency Metric Dimension of Some Families of Graph

Ali N. A. Koam,¹Ali Ahmad,²Muhammad Azeem,³Adnan Khalil,⁴and Muhammad Faisal Nadeem⁵

Academic Editor: Muhammad Gulzar

Received27 Apr 2022

Accepted13 Jun 2022

Published25 Jun 2022

Abstract

Metric dimension of a graph is a well-studied concept. Recently, adjacency metric dimension of graph has been introduced. A set is considered to be an adjacency metric generator for if (supposing each pair); there must exist a vertex along with the condition that is indeed adjacent to one of . The minimum number of elements in adjacency metric generator is the adjacency metric dimension of , denoted by In this work, we compute exact values of the adjacency metric dimension of circulant graph , Möbius ladder, hexagonal Möbius ladder, and the ladder graph.

1. Introduction

Let be a simple connected graph with vertex set and edge set Let be a set of nonnegative integers; we assume a function defined as

Then, is a metric space. A subset is called a metric generator for if it is the generator of the metric space ; that is, every point of the space is uniquely determined by its distances from the elements of . A minimum metric generator is the metric basis, and its cardinality is the metric dimension of , denoted by ; for further detail of metric and their parameters, see [1–6]. The concept of metric dimension was first introduced by [7] in the problem of uniquely determining the location of an intruder in a network and was named as a locating set instead of metric generators. The same concept of metric generators or locating set was introduced by [8], and this time, they named it as resolving sets. Application of metric dimension to the navigation of robots in networks is studied in [9] and to chemistry in [10, 11]. This concept was further studied by many researchers (for instance, see [12, 13]); different studies of metric and its related parameters are studied in literature; for example, metric dimension of bilinear form graphs is found in [14], distance-regular graph is available in [15], in terms of dominating set [16], study of the corona product is found in [17], and some computer fields are attached to this topic found in [18, 19]. Several metric generators have since been developed and researched, including independent resolving sets [20], resolving dominant sets [21], strong resolving sets [22], and local metric sets [23].

A subset of the set of vertices is an adjacency metric generator of if for every two vertices , there exists a vertex such that is indeed adjacent to one of . The minimum number of elements in the adjacency metric generator is the adjacency metric dimension of and symbolized by The idea of adjacency metric dimension was put forward by [24], and it is closed related to -locating dominating set [25]. The goal of this definition is to look at the metric dimension of the lexicographic product of graphs in terms of graph adjacency. For two graphs with -orders, the researchers in [26, 27] had taken the corona product of both graphs; the resulted graph has (local) metric dimension times the (local) adjacency metric dimension. They demonstrated that calculating the adjacency metric dimension is NP-hard as a result of this tight relationship.

The adjacency metric generator of any graph can be a metric generator in a correctly selected metric space, as pointed out in [26, 27]. Assume is a positive number. Assume the is a distance function, which is defined as

Then, the metric dimension of is equal to the adjacency metric dimension of A subset is the adjacency metric generator for a graph , if for each pair of vertices , there are at least vertices as a result for each For the minimum number () of members in the adjacency metric generator, the definition will be called as adjacency metric dimension of a graph , and here, it is symbolized by

The adjacency metric dimension is an NP-hard problem [25, 28], and it is very important to determine its exact values for well-known families of graphs. The primary goal of this work is to determine the exact values of the adjacency metric dimension of particular graph families, notably circulant graphs , ladder, Möbius ladder, and hexagonal ladder graphs. To compute the adjacency metric dimension of these classes, we need the following proposition by Moreno et al. [29].

Proposition 1 (see [29]). If is a graph with , then if and only if and .

Remark 2 (see [29]). If is a graph with , then .

2. Construction of Graphs

Circulant graphs are a type of graph that may be utilised in the construction of local area networks. Let and be the nonnegative with given conditions, , where With the collection of vertices in an undirected graph and the set of edges , the indices are considered to be taken in modulo condition, called a circulant graph, and it is symbolized by The generators are -numbers, and the edge is of type . Actually, we can observe that the circulant graph is an -regular graph, and the given is

Möbius ladder is built by a grid of , and twist this grid at 180°; now, paste the extreme most left and right path of vertices as seen in Figure 1. It contains -horizontal cycles of order four. For more study on ladder-type networks, see [30, 31]. The metric dimension of is three [32].

Hexagonal Möbius ladder is built in [33], it can construct by dividing each horizontal edge of a square grid by inserting a new vertex, and it becomes a grid of with each cycle having order six; now, twist this grid at 180° and paste the extreme most left and right path of vertices as shown in Figure 2. This graph contains -horizontal cycles of order six. The metric dimension of the hexagonal Möbius ladder network is three [33].

Let be a ladder graph [34], with We label the ladder graph as shown in Figure 3. The order and size of the ladder graph are and , respectively.

3. Results on Adjacency Metric Dimension of Graphs

We compute the adjacency metric dimension of certain graph families in this section. Let be a circulant graph obtained from by joining all the vertices at distance Let ; then, and In the next theorem, we compute the exact value of the adjacency metric dimension of the circulant graph .

Theorem 3. Let be a circulant graph. Then,

Proof. We divide the proof of the theorem into three parts.
Part A. For the inequality , when , let as an adjacency metric generator; then, the representation of are as follows , and For any vertex , we have the following representation: where , when ; otherwise, Also, when and , when ; otherwise, both are
From the above discussion, we can say that
Converse: on the contrary, assume that , for Let be a resolving set for adjacency metric dimension. Then, for any , the distance Given , by the principle of Dirichlet’s box, two or more components of resulted in identical vectors of distance, and further, this implied as a contradiction. Now, for , let ; we have the same representations, that is, either or Let the adjacency metric generator ; then, we have the same representations, that is, either or Let ; then, we have the same representations, that is, , , or Therefore, , concluding that
Part B. When , we intend to use the induction method on the order of cycle; set the base step as , which implies that Let be a resolving set for adjacency metric dimension. All vertices have the following representations: where , when , respectively, and when , when , when , and when ; otherwise, all values of are All the representations are different so On the contrary, assume that On this contradiction, the following are some cases to defend it.

Case 1. Let the adjacency metric generator , taking vertices in with -size gap (consecutive vetices). Then, we have the same representations in ; it means that have the same representations with the vertices which are not adjacent to any of the vertex of

Case 2. Let , taking vertices in with 1- and 0-size gap, respectively. Then, we have the same representations in ; it means that have the same representations with the vertices which are adjacent to only.

Case 3. Let , , taking vertices again in with - and -size gap, respectively. Then, we have the same representations in ; it means that have the same representations with the vertices which are adjacent to only.

Case 4. Let , taking vertices in with -size gap. Then, we have the same representations in ; it means that have the same representations with the vertices which are adjacent to only.

Case 5. Let , with - and -size gap, respectively. Then, we have the same representations in , which simplifies that have the same representations with the vertices which are adjacent to only.

Case 6. Let , taking vertices in with - and -size gap, respectively. Then, we have the same representations in , which means that have the same representations with the vertices which are adjacent to only.
Due to the nature of the adjacency metric generator with three cardinalities, have the following gap size possibilities: (i)The first gap is even, and second is odd(ii)The first gap is odd and the next is even(iii)Both gaps are even(iv)Both gaps are odd

Case 7. Let , taking vertices in above any sizes of gap possibilities. Then, we have the same representations in either or only single vertex of adjacent to both vertices, which have the same representations. From all above cases, Hence, our base step is true: Now, assume that it is also true for , and we have to show that it is also true for which implies that Assume the assertion that Putting equations (8) and (9) in equation (10), we have where
Part C. Remaining cases when
Again, the method of induction can be used, and this implies that the base step becomes true for and as well ; for this purpose, let be a resolving set for adjacency metric dimension; all the vertices have the following representations: where when , respectively, and when , when , when , when , and when ; otherwise, all values of are On the contrary, assume that Let be the adjacency metric generator with cardinality having the following gap size possibilities. (i)All the gap sizes are even(ii)All the gap sizes are odd(iii)One gap size is odd, and others are even(iv)One gap size is even, and others are odd

Case 1. Let , , with any sizes of gap possibilities above defined. Then, we have the same representations in either or only single vertex of adjacent to both vertices, which have the same representations. From all above cases, Hence, Now, assume that it is also true for , and we have to show that it is also true for , which leads to the assertion that Assuming that Putting equations (14) and (15) in equation (16), we have where This completes the proof of the theorem with all possible cases on order of the graph.

Theorem 4. Let be a Möbius ladder graph. Then,

Proof. We divide the proof of the theorem into three parts.
Part A. Let be an adjacency metric generator, and those shown in Table 1 are the representations of all vertices for
All the vertices have different representations, and it proves that On the contrary, suppose that By using Proposition 1, it is not possible, concluding that Part B. When , this adjacency metric generator is which claims that All vertices have the following representations: where when , respectively, and when , when , when , and when ; otherwise, all are It proves the claim that ; on the other hand, suppose that , when

Case 1. Let , , taking vertices in with -size gap. Then, we have the same representations in and

Case 2. Let , with the vertices in with any size of gap discussed in Theorem 3, part 6 Then, we have the same representations in and It means that when choosing the resolving set for adjacency metric dimension with cardinality three and vertices with any gap size, there must exist two vertices with the same representations which have one belonging to -domain and the second from -domain. From all above cases, for the values of . Hence, Part C. Here, we will use the induction method. For the base step, we choose ; the having the adjacency metric generator , and the following are the representations of all vertices: where when , respectively, and when , when , when , and when ; otherwise, all values of are It concludes that The contradiction side gives us For this contradiction, possible cases are discussed in the converse of part b of the theorem. One can evaluate from the discussion that with cardinality three is not possible which implies that Now, assume that it is also true for , and we have to show that which leads to the inductive step: Assume that Using equations (24) and (25) in equation (26), we have It completes the proof of the theorem with all possible cases on -cycles of the Möbius ladder graph.

Theorem 5. Let be a hexagonal Möbius ladder graph. Then,

Proof. We break the proof of the theorem into four parts according to the adjacency metric dimension.
Part A. In this part, we claim that For the case , let the adjacency metric generator be , and those shown in Table 2 are the representations of all vertices of with respect to
In Table 2, the given representations are different, and it proves that ; on the reverse, for inequality which is , using Proposition 1 is not true, concluding that Part B. This part contains the adjacency metric dimension of the hexagonal Möbius ladder graph with ; for this, let the adjacency metric generator be , and those shown in Table 3 are the representations of all vertices according to , which are again different, and it proves that ; on the reverse, inequality which is is not true, and the following are some discussions to this side.

Case 1. Let , taking vertices in with any size of gap discussed in Theorem 3, part 6 Then, we have the same representations in either or and It means that when choosing the resolving set of adjacency metric dimension with cardinality three and vertices with any gap size, there must exist two vertices with the same representations which have one belonging to -domain and the second from -domain. It is also possible that the same representations are in two vertices, and is one of them, and the second vertex belongs to -domain. From all above cases, Hence, Part C. For , let the adjacency metric generator be , and representations are shown in Table 4.

The representations given in Table 4 prove that On contrary, suppose that ; the following are some discussions to the side of this contradiction.

Case 1. Let , taking vertices in with any size of gap already discussed above in the theorem. Then, we have the same representations in either or and It means that when choosing the resolving set for adjacency metric dimension with cardinality five and vertices with any gap size, there must exist two vertices with the same representations which have one belonging to -domain and the second from -domain. It is also possible that the same representations are in two vertices and is one of them, and the second vertex belongs to -domain. From all above cases, Hence, Part D. In this part for the proof of where , we will apply the induction method and the base case for , which implies that , to prove that ; the following are the representations with respect to the adjacency metric generator : where when and when , when , when , when , when , when , when , and when ; otherwise, all values of are All the representations show that there are no two vertices with the same representation; hence, On the contrary, suppose that ; the following are some discussions to the side of contradiction.

Case 1. Let and , taking vertices in with any size of gap discussed in Theorem 3, part 6 Then, we have the same representations in either or or and It means that when choosing the resolving set of adjacency metric dimension with cardinality seven and vertices with any gap size, there must exist two vertices with the same representations which have one belonging to -domain and the second from -domain. It is also possible that the same representations are in two vertices and is one of them, and the second vertex belongs to -domain. From all above cases, Hence, Now, assume that the assertion is true for : We have to show that it is also true for Suppose Plugging the values of equations (34) and (35) in equation (36), we have As a result, the result holds for all positive integers It also completes the all possible cases that we assume on the very start of the proof.

Theorem 6. Let be a ladder graph with . Then,

Proof. We prove that with the induction method and the base step is which implies that If we assume on the contrary it is not possible that , for , all vertices have the following representations with respect to : where when , respectively, and when , when ; otherwise, all values of are All the representations show that there are no two vertices with the same representation; hence, Now, assume that the assertion is true for : We have to show that it is also true for Suppose Using equations (40) and (41) in equation (42), we have where As a consequence, the result holds for all positive integers for , which completes the proof.

4. Conclusion

In this article, we studied the adjacency metric dimension of circulant, ladder, Möbius ladder, and hexagonal Möbius ladder graphs. It is known that adjacency metric dimension is a useful parameter in localization, networking, and some robot navigation ideas. Therefore, it is interesting to find adjacency metric generators for generalized classes of graphs. We demonstrated that in adjacency metric generators, all graphs have inconstant numbers of members and that each graph follows the modification of parameters or order.

Data Availability

There is no data available.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2022 Ali N. A. Koam et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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