Journal of Chemistry

Volume 2019, Article ID 9259032, 9 pages

https://doi.org/10.1155/2019/9259032

## On Resolvability Parameters of Some Wheel-Related Graphs

^{1}Department of Computer Science and Technology, Hefei University, Hefei 230601, Anhui, China^{2}Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Lahore, Pakistan

Correspondence should be addressed to Hafiz Muhammad Afzal Siddiqui; moc.liamg@iuqiddisamh

Received 11 July 2019; Accepted 12 October 2019; Published 28 November 2019

Guest Editor: Shaohui Wang

Copyright © 2019 Bin Yang 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.

#### Abstract

Let be a simple connected graph, be a vertex, and be an edge. The distance between the vertex and edge *e* is given by , A vertex distinguishes two edges , if . A set *S* is said to be resolving if every pair of edges of *G* is distinguished by some vertices of *S*. A resolving set with minimum cardinality is the basis for *G*, and this cardinality is the edge metric dimension of *G*, denoted by . It has already been proved that the edge metric dimension is an NP-hard problem. The main objective of this article is to study the edge metric dimension of some families of wheel-related graphs and prove that these families have unbounded edge metric dimension. Moreover, the results are compared with the metric dimension of these graphs.

#### 1. Introduction and Preliminary

The molecular structure of a chemical compound is known as molecular graph (or chemical graph). There are different structures of chemical compounds, and every structure correlated with many chemical properties, and these properties can be calculated through some specific mathematical formulas. Graph theory helps us to study and analyze these dense structures in detail. In graph theory, we consider every atom as a vertex and covalent bound between atoms as an edge. A *graph G* consists of a set of objects called vertices, set of connections between vertices called edges , and graph *G* is usually denoted as . The number of vertices in *G* denoted by is often called the order of *G*, while the number of edges is known as size of *G* and denoted by . The concept of metric dimension was introduced by Slater in 1975 [1], and this concept independently was elaborated for graphs by Harary and Melter in 1976 [2]. After these two papers, lot of research papers have been published related to its theoretical properties and applications. The theoretical studies on metric dimension are highly contributed by several authors like, for instance, independent resolving sets [3], resolving dominating sets [4], strong resolving sets [5], local resolving sets [6], *k*-metric generators [7, 8], simultaneous metric generators [9], resolving partitions [10], strong resolving partitions [11], and *k*-antiresolving sets [12]. The metric dimension has many applications in the digital world and in mathematical chemistry such as network verification [13], robot navigation, image processing and pattern recognition [14], mastermind game [15], chemistry [16, 17, 18], and pharmaceutical chemistry [18].

If *G* is a simple connected graph and *u* and are any two vertices of *G*, then the distance is the length of a shortest path between *u* and . Let be an ordered set of vertices of *G* and let be a vertex of *G*. The representation of with respect to *S* is the ordered *k*-tuple . If , then we shall say that vertex distinguishes two vertices *x* and *y*. If distinct vertices of *G* have distinct representations with respect to *S*, then *S* is called a resolving set for *G*. A resolving set of minimum cardinality is called a basis for *G* and the cardinality is known as metric dimension of *G*, and it is denoted as .

Let *S* be a set of vertices of a simple connected graph *G*. Then, *S* is called an edge metric generator for *G* if every two edges of *G* are distinguished by some vertices of *S*. The minimum cardinality of *S* is called the edge metric dimension, and it is denoted by . For an ordered set of vertices of a graph *G* and any edge *e* in *G*, we refer to the *k*-vector (ordered *k*-tuple) as the edge metric representation of *e* with respect to *S*. The set *S* is an edge metric generator for *G* if and only if for every pair of different edges , of *G*, .

Let be a family of connected graphs of order for which . If there exists a constant such that for every , then we shall say that has a bounded edge metric dimension, otherwise has unbounded edge metric dimension. Moreover, if , , then has a constant edge metric dimension.

There is a controversy in the definition of edge resolving set and edge metric dimension. In the literature, another version of edge metric dimension exists, which is based on the distance between edges, denoted as . This version of edge metric dimension follows the following steps. *Step 1*. Convert a graph *G* into a line graph . *Step 2*. Find metric dimension, of line graph .

According to the above steps, , for reference see [19]. But in our case, the distance is based on vertex to edge, , as defined by Kelenc et al. [20].

In 1979, Garey and Johnson [21] pointed out that the finding of the metric dimension of a graph is an NP-hard problem. In 1994, Khuller et al. [22] also showed by another construction that the metric dimension of a graph is an NP-hard problem.

Recently, Kelenc et al. [20] have introduced the notion of edge metric dimension of graph, denoted by , and they presented some results relating the notions of metric dimension and edge metric dimension for some families of graph such as for path : where , for cycle : , for complete graph : , where , for complete bipartite graph : , where , for tree which is not a path: , for grid graph (where ): , for wheel graph (): , and for the grid graph : . They also proved that the edge metric dimension is an NP-hard problem. Moreover, they have raised many open problems related to edge metric dimension. Furthermore, in 2019, Zubrilina has classified some graphs on the same topic which has the in her paper [23]. Recently, in 2019, Rafiullah et al. studied the edge metric dimensions of wheel-related convex polytopes in [24] and characterized these graphs.

This paper is based on the question raised in [20] to characterize the families of graphs which observe one of the relations , , or .

We consider Jahangir graph , helm graph , sunflower graph , and friendship graph for edge metric dimension (NP-hard problem). Our aim is to characterize these families with respect to the nature of edge metric dimension. Moreover, we present closed formula__s__ for edge metric dimension of these graphs. For further reading on metric dimension and edge metric dimension, we refer [19, 24, 25, 26, 27].

#### 2. Main Results

##### 2.1. Jahangir Graph ()

The *Jahangir graph * obtained from wheel graph by alternately deleting *n* spokes. The order of is , and the size is . The Jahangir graph is also referred to as gear graph [28]. The metric dimension of Jahangir graph is given in [29]. Now, we present some observations for in the form of lemmas which help us to compute its edge metric dimension.

Lemma 1. *The central vertex does not belong to any edge metric basis S of , for .*

*Proof. *Let be a gear graph of order . The vertices of are labeled as , and edges of are labeled as and for all and as shown in Figure 1. The vertices and are named as major and minor vertices, respectively. Since and , does not belong to any basis of .

Let be a basis of and number of vertices on for . We say that the pairs of vertices for and are pair of neighboring vertices. We define the gap for as the set of vertices and . The number of gaps is *r* and some of which may be empty. If or , then the gaps and are known as the neighboring gaps. Let and or and with . Then, the gap between *u* and is denoted by . The number of vertices between *u* and is known as the size of gap. There are three types of gaps observed on of , which are , , and . We represent the major vertices by in the following proofs.