On the Geodesic Identification of Vertices in Convex Plane Graphs

Alsaadi, Fawaz E.; Salman, Muhammad; Rehman, Masood Ur; Khan, Abdul Rauf; Cao, Jinde; Alassafi, Madini Obad

doi:https://doi.org/10.1155/2020/7483291

Mathematical Problems in Engineering

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Volume 2020 | Article ID 7483291 | https://doi.org/10.1155/2020/7483291

On the Geodesic Identification of Vertices in Convex Plane Graphs

Fawaz E. Alsaadi,¹Muhammad Salman,²Masood Ur Rehman,³Abdul Rauf Khan,⁴Jinde Cao,^5,6and Madini Obad Alassafi¹

Academic Editor: Muhammad Javaid

Received20 Aug 2020

Accepted01 Dec 2020

Published28 Dec 2020

Abstract

A shortest path between two vertices and in a connected graph is a geodesic. A vertex of performs the geodesic identification for the vertices in a pair if either belongs to a geodesic or belongs to a geodesic. The minimum number of vertices performing the geodesic identification for each pair of vertices in is called the strong metric dimension of . In this paper, we solve the strong metric dimension problem for three convex plane graphs by performing the geodesic identification of their vertices.

1. Introduction

The identification of vertices of a graph using various graph parameters is a fascinating problem for researchers. In the literature, almost over 1000 articles are contributed to explore the identification of vertices in a remarkable way by using various graph theory concepts including graph coloring, labeling of vertices, domination in graphs, vertex covering, graph automorphisms with symmetry breaking technique, independence of vertices, and by defining the metric on graphs, to name a few.

By defining the metric on a graph, the problem of identification of vertices attracted many researchers due to its significant applications in several extents including verification, security, and discovery of networks [1], the chemistry of pharmaceutics for drug designing [2], mastermind game strategies [3], navigation of robots [4], connected joins in graphs [5], and solution of coin weighing problems [6]. Because of these practical significances of this problem, from the last two decades, numerous researchers identified vertices of a graph by considering the metric-related well-known concept of the metric dimension [2, 7, 8]. Later on, many researchers extended the study of this concept by defining its several variations including the fractional metric dimension [9], the resolving domination [10], the doubly metric dimension [11], the independent metric dimension [12], the weighted metric dimension [13], the -metric dimension [14], the solid metric dimension [15], the mixed metric dimension [16], the local metric dimension [17], the simultaneous metric dimension [18], the strong metric dimension [5], and the connected metric dimension [19].

This paper is aimed to identify the vertices of three convex plane graphs by using geodesics between them, which provides the strong metric dimension of these graphs.

2. Geodesic Identification: The Strong Metric Dimension

Let be a simple connected graph. A shortest path between two vertices and in is known as a geodesic. The geodesic identification of vertices is equivalently the strong resolvability of vertices, which is defined as follows: a vertex of performs the geodesic identification for the vertices in a pair (, strongly resolves the vertices in a pair ) if either belongs to (lies on) a geodesic or belongs to (lies on) a geodesic. A set of vertices in is said to be a strong metric generator for if for every pair of vertices of , there is always a vertex in which performs the geodesic identification for the vertices in the pair. The cardinality of such a smallest set is called the strong metric dimension of , denoted by [5].

Sebö and Tannier initiated the study of geodesic identification of vertices by defining the concepts of strong metric generator and strong metric dimension in 2004 [5]. Later on, this study was extended by many researchers, and they contributed to the literature with a variety of remarkable research work. To develop the readers interest, we shortly survey the strong metric dimension problem as follows:(i)The strong metric dimension problem has been solved for Sierpiński graph in [20], for hamming graphs in [21], for some convex polytopes in [22, 23], for wheel related graphs (including -fold wheel, sunflower, helm, and friendship graphs) in [24], for path, cycle, complete, complete bipartite, and tree graphs in [25], for Cayley graphs in [26], for Cartesian sum graphs in [27], for the power graph of a finite group in [28], for distance-hereditary graphs in [29], for generalized butterfly and starbarbell graphs in [30], for antiprism and king graphs in [31], for sun, windmill, and Möbius ladder graphs in [32], and for crossed prism in [33].(ii)The strong metric dimension of various products of graphs including Cartesian product, direct product, strong product, lexicographic product, rooted product, and corona product has been supplied through the articles in [26, 27, 30, 31, 33–39].(iii)The fractional version of the strong metric dimension problem has been introduced in [40] and further studied for various graphs and graph products in [41, 42].(iv)The technique of the computation of strong metric dimension with the concept of the vertex cover number has been provided in [38] by proposing the construction of strong resolving graph of a connected graph. Furthermore, the article in [25] supplied some fundamental realizations and characterizations of the strong metric dimension problem in connection with the strong resolving graph.(v)To solve the strong metric dimension problem, the genetic algorithmic approach is used in [43], and the variable neighborhood search method is used in [44].(vi)The article in [45] supplied the Nordhaus–Gaddum type results whenever the strong metric dimension problem was solved for graphs and their compliments.(vii)To compute the strong metric dimension of graphs using optimization techniques, the integer linear programming model for the strong metric dimension problem was formulated in [22].(viii)The complexity and the optimal approximability in the computation of the strong metric dimension problem of graphs have been discussed in [38, 46].(ix)Furthermore, we refer a survey in [47] and the Ph.D. thesis in [48] to the readers having interest in the computation of the strong metric dimension of graphs.

With this paper, we extend the study of the identification of vertices using the concept of strong metric dimension. We consider three convex plane graphs and investigate their strong metric dimension by performing the geodesic identification of vertices in the graphs.

3. Basic Works

Let be a simple and connected graph with vertex set and edge set . We denote the adjacent vertices and by and nonadjacent vertices by in . The neighborhood of a vertex in is . The number of vertices adjacent with a vertex is called its degree and is denoted by . The metric on is a mapping defined by , where is the length of (the number of edges in) a geodesic. Accordingly, a vertex of performs the geodesic identification for the vertices in a pair if and only if either .

Kratica et al. in [22] supplied the following two results, which are useful tools for the geodesic identification of vertices.

Lemma 1. (see [22]). Let be a pair of distinct vertices in a connected graph such that

Then, there is no vertex in which performs the geodesic identification for the pair .

Proposition 1. (see [22]). If is a strong metric generator for a connected graphs , then for every pair of distinct vertices in satisfying both the conditions in (1), either or .
For a connected graph , the number is called the diameter of , where is the eccentricity of a vertex . The following result, provided by Kratica et al. in [22], is also a useful tool for the geodesic identification of vertices.

Proposition 2. (see [22]). If is a strong metric generator for a connected graphs , then for every pair of distinct vertices in such that , either or .
A vertex of for which is known as a leaf. The following result describes the geodesic identification of leaves in a connected graph.

Lemma 2. If is a strong metric generator for a connected graphs and is a pair of distinct leaves in , then either or .

Proof. Since and are leaves, . Without loss of generality, let and . Now, if , thenIf , thenThe identities (2)–(5) provide that the pair satisfies both the conditions in (1), and hence, Proposition 1 yields the required result.
Lemma 2 supplies the following straightforward proposition.

Proposition 3. If is a strong metric generator for a connected graph and is the set of leaves in , then must contain at least leaves from .

4. Convex Plane Graphs

A graph is convex if a straight line joining any two vertices of entirely lies within the region occupied by . A graph is a plane graph if it is free from crossing of edges. Throughout this section, it should be helpful to keep in mind the following points:(i)The right arrow in the supper script of the notation of geodesics will indicate forward nature of geodesics (see Figure 1).(ii)The left arrow in the supper script of the notation of geodesics will indicate backward nature of geodesics (see Figure 1).(iii)The indices greater than or less than will be taken modulo .

In the next sections, we investigate the strong metric dimension of three convex plane graphs.

4.1. Convex Plane Graph

The graph of a convex polytope is defined in [49] for . The vertex set of is , and its edge set is

The convex plane graph ( for pendant) can be obtained from the graph of a convex polytope by attaching one pendant vertex (leaf) to the vertex of for each (see Figure 2) [50]. Thus, the vertex and edge sets of are

We investigate the strong matrix dimension of by supplying the following main result.

Theorem 1. For , let be a convex plane graph. Then,

The next four lemmas will lead the proof of Theorem 1.

Lemma 3. For odd values of , if is a strong metric generator for , then .

Proof. Let with , and consider Table 1 for .
In , note the following points: P1: is the set of leaves. P2: for each , . It follows, according to Table 1, that the pair satisfies both the conditions in (1). P3: for each .From P1, without loss of generality, we suppose that by Proposition 3. Then, since the pair satisfies P2 and P3, we have either or , by Propositions 1 and 2. Hence, .

Lemma 4. For even values of , if is a strong metric generator for , then .

Proof. Let with , and consider Table 2 for .
In , the following points hold: P1: is the set of leaves. P2: for each , . Therefore, Table 2 yields that the pairs and satisfy both the conditions in (1) for each . Table 2 further implies that the pair satisfies both the conditions in (1) because for each . P3: for each .Due to P1, without loss of generality, we let , by Proposition 3. Then, as the pairs and satisfy P2 and P3, we must have either or or or , by Propositions 1 and 2. For the geodesic identification of these three pairs, it is enough to consider in . Furthermore, for each , either or , due to P2 and Proposition 1. Hence, .

Lemma 5. If with , then the set is a strong metric generator for .

Proof. For any and any , since the vertex performs the geodesic identification for the pair , we have to perform the geodesic identification for every pair of distinct vertices with . For each , consider the following geodesics of length :Each geodesic, listed from (9) to (16), provides that the vertex performs the geodesic identification for every pair with . It follows that is a strong metric generator for .

Lemma 6. If with , then the set is a strong metric generator for .

Proof. We have to perform the geodesic identification each pair of distinct vertices for because for any and any , the pair possesses the geodesic identification by the vertex . For each , consider the following geodesics and geodesics of length :Except the pairs for and , all the pairs of vertices owned the geodesic identification by the vertices and due to the geodesics listed from (17) to (24). By considering the following geodesicwe get that the vertex performs the geodesic identification for all the remaining pairs . Hence, is a strong metric generator for .

Proof. (Theorem 1). The lower bound for the strong metric dimension of is provided by Lemmas 3 and 4, whereas the upper bound is due to Lemmas 5 and 6.

4.2. Convex Plane Graph

The graph of a convex polytope is defined in [49] for . The vertex set of is , and its edge set is

The convex plane graph ( for pendant) can be obtained from the graph of a convex polytope by attaching one pendant vertex (leaf) to the vertex of for each (see Figure 3) [50]. Accordingly, the vertex and edge sets of are

Mainly, we have the following result for the strong metric dimension of .

Theorem 2. For , let be a convex plane graph. Then,

The proof of Theorem 2 will be followed after the next four lemmas.

Lemma 7. For odd values of , if is a strong metric generator for , then .

Proof. Let with , and consider Table 3 for .
Note the following three points in : P1: is the set of leaves. P2: and for each . Hence, Table 3 ensures that pairs of vertices in satisfying both the conditions in (1) are , and for each . P3: for each .By Proposition 3, let us take without loss of generality due to P1. Then, from the remaining pairs , and , we must have either or or or , by Propositions 1 and 2, because these pairs are satisfying P2 and P3. Here, it can be seen, by letting only, that all these pairs possess the geodesic identification by the vertex . Furthermore, for each , from the pair satisfying P2, either or , by Proposition 1. Hence, .

Lemma 8. For even values of , if is a strong metric generator for , then .

Proof. Let with , and consider Table 4 for .
The following points are considerable in : P1: is the set of leaves. P2: and , which implies according to Table 4, that the pairs and satisfy both the conditions in (1) for each . Moreover, for each , , and so Table 4 concludes that the pair also satisfies both the conditions in (1). P3: for all .Now, without loss of generality, let us take in , by Proposition 3 and due to P1. Then, from the remaining pairs and satisfying P2 and P3, respectively, we must have either or by Proposition 2, and either or by Proposition 1. If we take in , then this vertex performs the geodesic identification for both the remaining pairs. Also, Proposition 1 yields that either or for each and either or for each because of the point P2. Therefore, .

Lemma 9. For with , the set is a strong metric generator for .

Proof. Note that every pair with and possesses the geodesic identification by the vertex . It follows that we should perform the geodesic identification for every pair of distinct vertices by the vertices in whenever . For each , except the pairs of vertices,All other pairs of vertices possess the geodesic identification by the vertices and using geodesics and geodesics of length , listed from the following equations:Now, for each , by considering geodesics, listed from (38) to (43), the vertex performs the geodesic identification for all the pairs of vertices listed in (29):It concludes that is a strong metric generator for .

Lemma 10. If with , then the setis a strong metric generator for .

Proof. Since the vertex performs the geodesic identification for every pair of vertices including , it implies that we have to perform the geodesic identification by the vertices in for all the pairs of vertices from . For this, consider the following geodesics of length :Due to geodesics, listed from (45) to (50), the vertex performs the geodesic identification for all the pairs of vertices except the pairs for all and the pair for each . These pairs are identified as follows:(i)For each , the vertex performs the geodesic identification for the pair due to the geodesic .(ii)For each , the vertex performs the geodesic identification for the pair due to the geodesic .(iii)For each , the vertex performs the geodesic identification for the pair due to the geodesic .(iv)For each , the vertex performs the geodesic identification for the pair due to the geodesic .Hence, we conclude that is a strong metric generator for .

Proof. (Theorem 2). The lower bound for the strong metric dimension of is provided by Lemmas 7 and 8, whereas the upper bound is due to Lemmas 9 and 10.

4.3. Convex Plane Graph

The graph of a convex polytope is defined in [49] for . The vertex set of is , and its edge set is

The convex plane graph ( for pendant) is obtained from the graph of a convex polytope by attaching one pendant vertex (leaf) to the vertex of for each (see Figure 4) [50]. Thus, we have the followings vertex and edge sets for :

The following main result provides the strong metric dimension of .

Theorem 3. For , let be a convex plane graph. Then,

The proof of Theorem 3 will be followed after the next four lemmas.

Lemma 11. For odd values of , if is a strong metric generator for , then .

Proof. Let with , and consider Table 5 for .
Note the following points in : P1: is the set of leaves. P2: for each , the pairs of vertices and satisfy both the conditions in (1) because of Table 5 and since , . P3: for each .Due to P1 and by Proposition 3, we let without loss of generality. Then, the remaining pair satisfies both P2 and P3. So, we must have either or , by Propositions 1 and 2. Furthermore, Proposition 1 yields that, from the pair , either or for all due to P2. Hence, .

Lemma 12. For even values of , if is a strong metric generator for , then .

Proof. Let with , and consider Table 6 for .
The following points hold in : P1: is the set of leaves. P2: for each , and . It follows, by Table 6, that the pairs of vertices , , , , and satisfy both the conditions in (1) for all . Moreover, for each , . Therefore, Table 6 again yields that the pairs of vertices and satisfy both the conditions in (1) for all . P3: for all .Without loss of generality, taking in , by Proposition 3 and due to P1, we get three pairs of vertices , and , which are still satisfying P2 and P3. For these pairs, Propositions 1 and 2 allow us to take either or or . It can be seen that the best suitable choice, to perform the geodesic identification for these three pairs, is to take in . For otherwise, we will get . Furthermore, the point P3 together with Proposition 2 yields that, from the pairs of vertices and , either or and or for all . Accordingly, we discuss the following four cases:(i)If and for all , then we must have either or for all , by the point P2 and Proposition 1. For otherwise, we get the pair of vertices , for all , which is left unidentified by the elements of , but we have .(ii)If and for all , then we must have either or for all , by the point P2 and Proposition 1. Otherwise, we get the pair of vertices , for all , which is left unidentified by the elements of , but we have .(iii)If , then of course for all . Moreover, we must have either or for all , by the point P2 and Proposition 1. For otherwise, we get the pair of vertices , for all , which is left unidentified by the elements of . Then, we have .(iv)If and for all , then we have .It can be concluded from these four cases that the most suitable choice to construct a strong metric generator with minimum cardinality is found in the Case 3, which yields that .

Lemma 13. If with , then the set is a strong metric generator for .

Proof. It is enough to perform the geodesic identification for those pairs of vertices of having no element from the set because every pair of vertices having one element from possesses the geodesic identification by the element . Let us consider the following geodesics of length , for each :The vertex performs the geodesic identification for all the pairs of vertices due to geodesics listed from (54) to (59), except the following pairs of vertices:For , , the following geodesics ensure the geodesic identification of the pairs, given in (60) and (61), by some vertices of as described in Table 7:Hence, we conclude that is a strong metric generator for .

Lemma 14. If with , then the setis a strong metric generator for .

Proof. For , a pair of vertices with possesses the geodesic identification by the vertex . Therefore, we should perform the geodesic identification for each pair of vertices of by the vertices in for both . For each , the following and geodesics of length are useful for our purpose:It can be seen that, for some pairs of vertices, the vertex performs the geodesic identification due to geodesics in (67)–(69), and for some pairs of vertices, the vertex performs the geodesic identification due to geodesics in (70)–(72). The pairs of vertices, which are left unidentified, are as follows:All these pairs of vertices possess the geodesic identification, as described in Table 8, with the help of the following geodesics, for , , , , and :Thus, is a strong metric generator for .

Proof. (Theorem 3). The lower bound for the strong metric dimension of is provided by Lemmas 11 and 12, whereas the upper bound is due to Lemmas 13 and 14.

5. Concluding Remarks

The geodesic identification (strong resolvability) for each pair of vertices has been performed in three families , , and of convex plane graphs for all . In fact, we extended the study of the identification of vertices by solving the strong metric dimension problem for three more families of graphs. The problem is solved with the following investigations:(i)In total pairs of vertices are identified with vertices by using geodesics when is odd and with vertices by using geodesics when is even.(ii)In total pairs of vertices are identified with vertices by using geodesics when is odd and with vertices by using geodesics when is even.(iii)In , total pairs of vertices are identified with vertices by using geodesics when is odd and with vertices by using geodesics when is even.

Data Availability

The data used to support the findings of this study are available from the corresponding author.

Conflicts of Interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.

Acknowledgments

This project was funded by the Deanship of Scientific Research (DSR) at King Abdulaziz University, Jeddah, under grant no. RG-19-135-38. The authors therefore acknowledge with thanks DSR for technical and financial support.

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Copyright © 2020 Fawaz E. Alsaadi 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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