Abstract
The number of edges in a shortest walk (without repetition of vertices) from one vertex to another vertex of a connected graph is known as the distance between them. For a vertex and an edge in , the minimum number from distances of with and is said to be the distance between and . A vertex is said to distinguish (resolves) two distinct edges and if the distance between and is different from the distance between and . A set of vertices in a connected graph is an edge metric generator for if every two edges of are distinguished by some vertex in . The number of vertices in such a smallest set is known as the edge metric dimension of . In this article, we solve the edge metric dimension problem for certain classes of planar graphs.
1. Introduction
In graph theory, the metric dimension problem (the problem statement and its background can be found in [1–3]) captivated numerous graph theorists because of its implications in various fields, which includes discovery of networks with security and verification [4], chemistry pharmaceutics in connection with the designing of drug [1], strategies of mastermind games [5], robot navigation [6], problems of coin weighing with its solutions [7], and connected joins in graphs [8]. Due to the empirical significance of this problem, for the last two decades, many researchers have tried to solve this problem by defining several versions of the metric dimension problem such as the fractional version [9], resolving domination [10], the version of doubly metric dimension [11], independent version [12], weighted version [13], the version of metric dimension [14], solid version [15], mixed version [16], edge version [17], local version [18], simultaneous version [19], connected version [20], and the strong version [8].
This paper is aimed to solve the edge metric dimension problem for five classes of planar graphs (class of convex polytopes), (class of chordal ring networks), (class of -graphs), (classes of planar graph), and (classes of planar graph) by investigating their edge metric generators.
2. The Edge Metric Dimension
All graphs mentioned in this paper are finite, connected, undirected, and simple. Moreover, the notations and will be used for vertex and edge sets of , respectively. The length (number of edges) of a shortest path between two vertices and in is known as the distance, , between them. Let be a vertex and be an edge in ; then, the distance between and is . The vertex distinguishes two distinct edges and of if . The metric code of an edge with respect to a set is the -vector:
The set is an edge metric generator for if for every two edges and of , we have . The cardinality of such a smallest set for is called the edge metric dimension of , denoted by [17].
Let be a class of connected graphs of order . If as for some , then we say that the edge metric dimension of is bounded. Otherwise, it is unbounded. The class of graphs is said to have a constant edge metric dimension if the edge metric dimension of graphs in the class is bounded and remains unchanged with the increase in the number of vertices of graphs.
The edge metric dimension problem was introduced by Kelenc et al. in 2018 [17]. They solved this problem for cycle graphs, trees, grid graphs, and complete bipartite graphs. Moreover, examples for the relationships , , and were also constructed by them, where the number denoted the metric dimension of [1]. Further, they explored that the determination of the edge metric dimension for a graph is an NP-complete problem. Later on, this investigation was stretched out by numerous scientists and they added to the writing with an assortment of momentous research work. For readers’ interest, we develop a short survey of results about this problem as follows:(i)Whenever we talk about the ratio between the edge metric dimension and the metric dimension of a graph , we found an example from [17] which illustrates that this ratio is approximately equal to , but this ratio cannot be bounded above in general as proved in [21].(ii)Graphs of order having the edge metric dimension are classified in [21], whereas graphs having the edge metric dimension are classified in [22].(ii)Relationships of edge metric dimension with the maximum and minimum degrees of a graph are developed in [17, 23], while the relationship of edge metric dimension with the clique number of a graph is established in [22].(iv)The edge metric dimension problem of various families and graph operations such as for the barycentric subdivision of Cayley graphs has been solved in [24]. Moreover, the edge metric dimension problem was solved for the join of graphs in [21, 25], for the Cartesian product of any graph with a path in [21], and for the lexicographic and corona products of graphs in [25].(v)Further, the edge metric dimension problem has been solved for various classes of graphs, such as for necklace graphs in [26], for two classes of generalized Peterson graphs and in [23], for web graphs, convex polytope , and prism-related graphs in [27], for sunlet graph and prism graph which turned out to be a constant in [28], for multiwheel graphs in [29], for two classes of circulant graphs and in [30], and for some classes of trees, namely, stars, brooms, double brooms, and banana trees, in [31].(vi)The edge metric dimension problem via hierarchical product and integer linear programming was discussed in [32].(vii)Graphs having the smaller edge metric dimension than the metric dimension have been explored in [33]. Further, it has been proved that it is not possible to bound the edge metric dimension of a graph by some constant factor of the metric dimension of [33].(viii)The local version of the edge metric dimension problem has been defined and studied for path graphs, ladder graphs, cycle graphs, star graphs, and wheel graphs in [34].
Kelenc et al. [17] and Filipovic et al. [23] supplied the following results, which are useful tools for the investigation of the edge dimension of graphs.
Proposition 1. (see [17]). Let be a connected graph and let be the maximum degree of . Then, .
Theorem 1. (see [23]). Let be a connected graph and let be the minimum degree of . Then, .
Corollary 1. (see [23]). Let be an regular graph. Then, .
3. Classes of Planar Graphs
In this section, we find the constant edge metric dimension of five classes of planar graphs, namely, family of convex polytopes, family of chordal ring networks, family of graphs, and two families of particular planar graphs.
3.1. Class of Convex Polytopes
The graph of convex polytope, , can be constructed with the combination of the graph of a convex polytope [35] and the graph of a prism [36]. The half planar view of this convex polytope is shown in Figure 1. The vertex set of is and the edge set iswhere the modulo will be used when the subcript is greater than .
Lemma 1. For with , the set is an edge metric generator.
Proof. Metric codes of the edges of with respect to the set are given as follows:It can be seen that every two distinct edges of have different metric codes with respect to , which implies that is an edge metric generator for .
Lemma 2. For with , the set is an edge metric generator.
Proof. Metric codes of the edges of with respect to the set are given as follows:It can be observed that for any two distinct edges and , . Therefore, is an edge metric generator for .
Theorem 2. For all
Proof. Note that the minimum degree in is Therefore, Theorem 2 implies that . Further, Lemma 4 and Lemma 5 supply that Hence,
3.2. Class of Chordal Ring Networks
A chord for a path in a graph is an edge of which joins the vertices of the path but not a part of that path. A chordal graph is an undirected graph whose every -cycle has a chord for . In the literature, numerous types of chordal ring networks can be found, but we consider chordal ring networks supplied by Arden and Lee [37]. These chordal ring networks can be obtained by adding chords in an even order cycle in a regular way. These chords connect an even numbered vertex to an odd numbered vertex [37, 38]. Mathematically, Arden and Lee’s purposed chordal ring networks are defined as follows: let a group of integers under addition module , and let three distinct odd elements , and from . Then, a chordal ring network of order , denoted by , is a graph whose vertices are labeled by the elements of , and the th labeled vertex will form an edge with the th, th, and th labeled vertices for each even . Accordingly, it can be seen that these chordal ring networks are cubic bipartite graphs since even numbered vertices are pairwise independent and so are the odd numbered vertices.
In this section, we consider chordal ring networks for , , and , whose one graph is shown in Figure 2. For simplicity, let us divide the vertex set of into two element sets as follows:
Accordingly, the edge set of should bewhere the modulo will be used on subscripts.
In [39], it has been proved that the family of chordal ring networks is planar whenever for The planar drawing of this family is shown in Figure 3.
It is an easy task to see that the set is an edge metric generator for . For and we have the following result.
Lemma 3. For with , the set is an edge metric generator.
Proof. Metric codes of the edges of with respect to the set are listed as follows:It can be noticed that every two distinct edges of have distinct metric codes with respect to the set . It follows that is an edge metric generator for .
Theorem 3. For with .
Proof. Since is a -regular graph, so , by Corollary 1. Further, Lemma 3 provides that Hence, .
3.3. Class of -Graphs
Each graph of this class is denoted by , which consists of copies of -graphs connected in a planar way as shown in Figure 4. Let the vertex set of be
Then, the edge set of is
Lemma 4. For , the set is an edge metric generator.
Proof. Metric codes of the edges of with respect to the set are given as follows:It can be seen that any two different edges of have different metric codes with respect to the set , which shows that is an edge metric generator for .
Theorem 4. For all ,
Proof. It can be seen that is a -regular graph. Therefore, Corollary 1 implies that . Further, Lemma 4 yields that Hence, we get the required result.
3.4. Classes of Planar Graph and
Planar drawings of graphs and are shown in Figures 5 and 6 for . For , the vertex set for both of these planar graphs has the same vertex set, which is
But their edge sets are
It is straightforward to see that the set is an edge metric generator for , which implies that . Further, Theorem 2 provides that because .
Lemma 5. For , the set is an edge metric generator.
Proof. Metric codes of the edges of with respect to the set are given as follows:It can be seen that every two distinct edges of have different metric codes with respect to the set , so is an edge metric generator for .
Lemma 6. For , if is a minimum edge metric generator for , then .
Proof. As , , by Theorem 2. Assume that and let . Without loss of generality, we let . Then, , and we have four cases to discuss.
Whenever for . Then,Whenever for . Then,Whenever for . Then,Whenever for . Then,In all these cases, we get a contradiction of the fact that is an edge metric generator for . Hence, our supposition is wrong and so .
Theorem 5. For , .
Proof Lemma. 5 and Lemma 6 supply the required result.
Lemma 7. For , the set is an edge metric generator.
Proof. Metric codes of the edges of with respect to the set are given as follows:It can be observed that every two distinct edges have different metric codes with respect to the set , so is an edge metric generator for .
Theorem 8. For , .
Proof. Note that the maximum degree in is . So, Proposition 1 implies that . Further, Lemma 7 yields that .
4. Conclusion
With this article, we have extended the study of the edge metric dimension problem towards few classes of planar graphs. We observed that in each of the considered classes of planar graphs, the edge metric dimension is not changed when we increase the number of vertices in graphs, which yields that the investigated edge metric dimension is constant. In fact, we proved that only three vertices are sufficient to distinguish (resolve) every pair of edges in each planar graph of the considered classes [40–44].
Data Availability
The data used to support the findings of the study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This research was supported by the Anhui Natural Science Research Project (2020) under grant no. KJ2020A0696.