Abstract
In this paper, a new concept -size edge resolving set for a connected graph in the context of resolvability of graphs is defined. Some properties and realizable results on -size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the -size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded -size edge metric dimension. Furthermore, the k-size edge metric dimension of the graphs Pm □ Pn, Pm □ Cn for m, n ≥ 3 and the generalized Petersen graph is determined. It is shown that these families of graphs have constant -size edge metric dimension.
1. Introduction
Kelenc et al. [1] recently defined the concept of edge resolvability in graphs and initiated the study of its mathematical properties. The edge metric dimension of graph is the minimum cardinality of edge resolving set, say , and is denoted as . An edge metric generator for of cardinality is an edge metric basis for [1]. This concept of an edge metric generator may have a weakness with respect to possible uniqueness of the edge identifying a pair of different vertices of the graph. Consider, for example, in a network, a vertex is identified by a unique edge in a metric basis , but if at some point the communication between the vertex and edge is blocked, then the vertex cannot be accessed by the edge metric basis . To avoid this situation, one can think of defining a metric edge basis in which every vertex can be identified by at least two edges. Inspired by the motivation of idea of -size resolving sets in graphs by Naeem et al. [2], we present a new concept in the context of edge resolvabililty, called the -size edge resolving set in graphs.
For an undirected, simple, and connected graph , the vertex set is and edge set is . The distance parameter in graphs has been used to distinguish (resolve or determine) the vertices or edges of . The distance between the vertex and the edge in a graph is given by . Any two edges and are resolved by a vertex of a graph , whenever . A set of vertices is an edge metric generator for a graph , whenever every two edges of are resolved by some vertex of . The edge metric dimension of graph is the minimum cardinality of set and is denoted as . An edge metric generator for of cardinality is an edge metric basis for [1].
Definition 1. A set of vertices is said to be a -size edge resolving set of a graph of order if is an edge resolving set and the size of subgraph induced by is equal to . The -size edge metric dimension of , denoted by , is the minimum cardinality of a -size edge resolving set of . Moreover, the -size edge resolving set of cardinality is represented as -set, where and belongs to a set of natural number.
Now, we discuss the existence of this new parameter in some simple, nontrivial connected graphs.
Let be a connected graph having the vertex set and edge set , as shown in Figure 1. A set is a 1ser-set for . It can be seen that, for , the -size edge resolving set for graph exist only for .
We observe that the set is a minimum -set for of Figure 1. Moreover, for the graph , -set does not exist.
We have the following remark from these two examples.
Remark 1. (i)The existence of does not imply the existence of for and vice versa in any nontrivial connected graph (ii), for any simple graph , where In this paper, we compute the -size edge metric dimension in several well known families of graphs, Cartesian product graphs , and generalized Petersen graphs . Moreover, we present some realizable result on -size edge metric dimension in graphs for .
2. Applications
Resolvability in graphs has diverse applications related to the navigation of robots in networks [3], pattern identification, and image processing. It has also many applications in pharmaceutical chemistry and drugs [4–6]. Few interesting connections between metric generators in graphs and the mastermind game or coin weighing problem have been presented in [7]. The other important results about the metric and edge metric dimension can be found in [8–12].
3. Existence of -Size Edge Resolving Sets in Well-Known Classes of Graphs
Now, we firstly initiate the study of existence of this new parameter in some basic families of graphs and compute their k-size edge metric dimension.
Lemma 1. For a path graph , if.
Proof. Consider a path graph with vertex set and edge set . Let be a subset of vertex set of . The code of each edge with respect to is distinct because each edge has the 0 entry at its and place. The code of each edge is , for , and , for . Thus, is an edge resolving set for . Since the size of is , it implies that the subgraph induced by has edges. Therefore, is -size edge resolving set for . Hence, for .
Lemma 2. For a simple and connected graph of order , if.
Proof. Let be a cycle graph of order , and let . We define . The code of each edge for and for with respect to has the 0 entry at its and place. The code of each edge is , for . The and entries of edges are equal to when is even and equal to and , respectively, when is odd . The code of remaining edges is , for , when is even and, for , when is odd. We note that the codes of all the edges are distinct. Therefore, is a -size edge resolving set for . Hence, for .
The -size edge resolving sets of a complete bipartite graph exist only for the values of given in the following result.
Lemma 3. For the complete bipartite graph and ,while, for , we have
Observation. It cannot be necessary that a -size edge resolving set has at least vertices in it.
To justify our above observation, we consider a graph . A set of vertices is the vertex set of . One can observe that the set is a 6-size edge resolving set for . Therefore, .
4. -Size Edge Metric Dimension of Cartesian Product of Graphs
Let be the Cartesian product of two path graphs and , for . Let be the set of horizontal edges and be the set of vertical edges of . The graph of is shown in Figure 2. To find distances, we embed into plane in such a way that each vertex is in an ordered pair form. Let the vertices be the corner vertices of . In the next two lemmas, we shall discuss size 1, size 2, and size 3 edge metric dimension of and , for .
Lemma 4. Let be the cartesian product graph ; then, we have
Proof. Here, we will prove this result for (only). For this we consider , where , and prove that is a -set for . Note that is the distance between any two vertices of . Let be an edge. The distances of the edge from the vertices of are calculated as follows.
, when , when , and when ; , whenever , , whenever , , whenever , , whenever , , whenever , , whenever , , whenever , and , whenever . Suppose contrary two edges and are at the same distance from the vertices of . Thus, we have the following equalities:The above equalities imply that . Thus, it follows that . In both the cases or , and we get and . The equality together with implies that . Both the vertices and can either equal to or equal . One of the edges or does not represent an edge if they have distinct values. So, finally we have , which is a contradiction. Therefore, is an edge resolving set for . Moreover, the subgraph induced by has 3 edges. Hence, we conclude that . Similarly, we can prove the result for the values of . Hence, we conclude the result.
We present the following result on the -size edge metric dimension of the Cartesian product graph without proof.
Lemma 5. Let be the cartesian product graph ; then, we have
5. -Size Edge Metric Dimension of Generalized Petersen Graphs
The generalized Petersen graph is a 3-regular graph containing vertices and edges. The vertex set of is , and the edge set is . The edges for are said to be spokes of . The outer cycle of is said to be the principal cycle .
We will compute the -size edge metric dimension of for and in the following two sections. Firstly, we will find upper bound for the 1-size edge metric dimension of .
Lemma 6. For all , we have .
Proof. Let be a set of vertices of .
Case 1. When is odd.
Here, we define . There is only one edge in the subgraph induced by and the codes of all the edges of are given in Tables 1–3.
Case 2. When is even.
We define . The induced subgraph by has only one edge, and the codes of all the edges of are given in Tables 4–6.We observe that codes of all the edges in both cases are distinct. So, is 1-size edge resolving set for . Hence, we have . Now, we will compute upper bound for the size 2 edge metric dimension of generalized Petersen graphs .
Lemma 7. For all ,.
Proof. Let be an edge resolving set for . Define . The codes of all the edges of with respect to are given Table 7–9.
For , the code of outer edge will be when is even and when is odd.
It seems that codes of all the edges are distinct. So, is size 2 edge resolving set for . Hence, we have .
In the next lemma, we will give the lower bound for the -size edge metric dimension of for .
Lemma 8. For all , we have .
Proof. First, we will show that there is no edge resolving set of consisting of two vertices. Contrarily, we suppose that be a set having two vertices of . Then, we have the following three possibilities.
Case 1. When both the vertices and are from the principal cycle.
Let us fix a vertex, say , then is any other vertex . For , we have . For , we have . For , we have when is even, while when is odd. For , we have when is even, while when is odd. For , we have .
Case 2. When both and are the inner vertices.
Let us fix a vertex, say , then is any other vertex .
For , we have . For , we have when is even, while when is odd. For , we have .
Case 3. When is any vertex from the principal cycle and is any inner vertex.
Let us fix a vertex, say ; then, is any inner vertex . For , we have . For , we have . For , we have when is odd; however, when is even. For , we have .
From the above three cases, we conclude thatSo, there does not exist a size 1 and size 2 edge resolving set of cardinality 2 in . Therefore, it yields that , for the value of .
Lemmas 6–8, we conclude the following main result.
Theorem 1. For all , we have when .
6. Bounds and Some Realizable Results on
From the earlier discussion, one fundamental question arises. Is the size edge metric dimension strictly greater than the -size metric dimension? To answer this question, we consider following two examples.
Consider the graphs and which are depicted in Figure 3. It can be observed that the set is an edge resolving set for and the cardinality of set is minimum. Moreover, is a -set, is a -set, and is a -set for . Thus, and .
While, for the graph , the set is an edge resolving set of the minimum cardinality. Here, the outer vertices are and the inner vertices are . It can be easily seen that the sets , , and are -set, -set, and -set, respectively, where , , and . Thus, and .
From these two examples, it can be observed that if exists for in a nontrivial connected graph of order , then
However, the following example shows that the above inequality is not true, in general.
Example 1. Let be a graph which is constructed from two graphs and . The vertex set of is and the edge set is , as shown in Figure 4. The set is a -set of . However, there is no such set which resolves all the edges of graph with cardinality 4 and has size 5. So, we take the set is a -set of the minimum cardinality for . Hence, .
Next, we characterize some realizable results for -set and -set in graphs.
Theorem 2. For a nontrivial, simple, and connected graph of order , we have if and only if .
Proof. Lemma 1 implies that if , then . Conversely, assume that be a connected graph of order and . Since the induced subgraph has only one edge, therefore it is obvious that . Thus, is a path graph of order 2.
The following result on the complete graph was presented in [1].
Lemma 9 (see [1]). For any integer , .
Theorem 3. Letbe a complete graph of order, thenexists iff. Moreover,.
Proof. One can observe that if is a complete graph , then the result holds. Conversely, let be a complete graph of order . Now, by Lemma 7, the induced subgraph has more than one edge. Therefore, does not exist. Hence, the proof is complete.
Theorem 4. Let be a nontrivial connected graph of order , then if and only if or .
Proof. Let or . From Lemmas 1 and 2, and Theorem 5, we have . Contrarily, assume that be a connected graph of order and . For , it is simple to prove that or . Now, we will prove the result for . For this, we suppose be a -set of of cardinality . It is easy to see that ; therefore, the induced subgraph has surely more than one edge in it ( is a connected graph). It yields that . Thus, or .
Remark 2. The two size edge resolving sets exist in complete bipartite graph if and only if . Moreover, , , and.
Theorem 5. For a simple and connected graphof order,if and only if.
Proof. By Lemma 1 and Remark 2, if , then . Conversely, suppose that for a connected graph of order . Let be a -set of order . Since is a connected graph and induced subgraph has two edges which implies that , thus .
Now, we study a sufficient condition for a pair of positive integers to be realizable as the order and the -size edge metric dimension of a connected graph, respectively.
Theorem 6. For a pair of positive integers with , there exists a connected graph of order and , where .
Proof. For , we consider the following two cases according to the choice of .(i)For , let be a path graph of order . Lemma 1 implies that , where . For , let be a cycle of order , then it seems that , where .(ii)For , let be a connected graph of order obtained from paths , where , and vertices with and for and , as shown in Figure 5. Firstly, we prove that . For this, let . Since, the subgraph induced by has edges and for each , it implies that is a -set for , and hence . Now, to prove , we suppose contrarily that and be a -set for with . If contains at least vertices from , say and the size of is equal to , it follows that . However, still we have . Thus, which yields that .
7. Conclusions
In this work, we have introduced a new variant, namely, the -size edge metric dimension of graphs, and initiated its study by finding the -size edge metric dimension of several well-known classes of graphs. We have characterized the graphs having -size edge metric dimension and . Moreover, we have computed the -size edge metric dimension of the Cartesian product graphs and for the values of . In addition, we have proved that the -size edge metric dimension of generalized Petersen graphs is 3 for the values of . Some realizable results on -size edge resolvability are also presented in this paper [8–12].
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.