# On the Edge Resolvability of Double Generalized Petersen Graphs

Authors: Tanveer Iqbal; Muhammad Rafiq; Muhammad Naeem Azhar; Muhammad Salman; Imran Khalid
Journal: Journal of Mathematics
 (2022)

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## Abstract

For a connected graphG=VG,EG, let v∈VG be a vertex and e=uw∈EG be an edge. The distance between the vertex v and the edge e is given by dGe,v=mindGu,v,dGw,v. A vertex w∈VG distinguishes two edges e1,e2∈EG if dGw,e1≠dGw,e2. A well-known graph invariant related to resolvability of graph edges, namely, the edge resolving set, is studied for a family of 3-regular graphs. A set S of vertices in a connected graph G is an edge metric generator for G if every two edges of G are distinguished by some vertex of S. The smallest cardinality of an edge metric generator for G is called the edge metric dimension and is denoted by βeG. As a main result, we investigate the minimum number of vertices which works as the edge metric generator of double generalized Petersen graphs, DGPn,1. We have proved that βeDGPn,1=4 when n≡0,1,3mod4 and βeDGPn,1=3 when n≡2mod4.


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## Body

## 1. Introduction

The concept of the resolving set or the locating set was introduced by Slater [1] and then by Harary and Melter [2] separately. A resolving set R of a given graph is the subset of the vertex set of the given graph so that the vertices of the graph can be located at a unique location by the distances of them from the vertices of R. The cardinality of a such a smallest set R for G is called the metric dimension of G, denoted by βG [1, 2]. Resolvability in graphs has diverse applications related to the navigation of robots in networks [3–5], pattern identification, and image processing. It has also many applications in pharmaceutical chemistry and drugs [6–8]. In [9], some interesting connections of metric generators of graphs with the mastermind game and coin weighing problem have been presented. This concept was used by Slater [1] to determine the location of an intruder in a network in a unique way. However, if there is an intruder that cannot be accessed by its nodes or vertices rather by using the links between them called edges, then, in this case, the location of an intruder in a network need not to be identified. Thus, in this situation, there is need to define extra condition to overcome the problem.For a simple, undirected, and connected graphG, the vertex set and edge set of G is denoted by VG and EG, respectively. The vertices and the edges of G can be resolved or determined with the help of distance parameter. The distance between a vertex α and an edge β=α1α2 in a graph G is given by dβ,α=mindα1,α,dα2,α. Any two edges β1 and β2 are said to be resolved (identified) by a vertex α of G if and only if dβ1,α=dβ2,α implies β1=β2. A set of vertices X⊆VG is an edge resolving set for G if and only if every two distinct edges of G are resolved by some vertex in X. The edge metric dimension of G is the cardinality of such a smallest set X, denoted by βeG [10].


## 2. Literature Review

Motivated by this, Kelenc et al. [10] recently defined the concept of edge resolvability in graphs and studied its different properties. In their work, they extended the study of the resolving set to the edge resolving set. There are several graphs for which the metric dimension is equal to the edge metric dimension, the metric dimension is less than the edge metric dimension, and the metric dimension is greater than the edge metric dimension [10]. Moreover, exact values for the edge metric dimension of many classes of graphs were computed, while the upper and lower bounds of many other graphs have been given. The results of the mixed metric dimensions and the edge metric dimensions for some families of graphs are given in [11, 12]. Peterin and Yero [13] computed exact formulae for different products of graphs. Zubrilina [14] categorized some properties of edge metric dimension of graphs. Moreover, we refer the reader to the work [15–19] where some related results on this topic can be found.


## 3. Double Generalized Petersen Graph

The double generalized Petersen graphDGPn,k is a cubic graph, for n≥3 and 1≤k≤n−1, with vertex set VDGPn,k=xi,ui,vi,yi:1≤i≤n and edge set EDGP=xixi+1,xiui,uivi+k,viui+k,viyi,yiyi+1:1≤i≤n. The exterior cycle of the DGPn,k contains vertices xi,1≤i≤n, in which every vertex xi is adjacent to the vertex ui to make spokes of type xiui:1≤i≤n. Similarly, the inner cycle of the DGPn,k contains the vertices yi,1≤i≤n, in which every vertex yi is adjacent to the vertex vi to make spokes of type viyi:1≤i≤n [20]. Throughout the study, the indices will be taken under modulo n. For n=13 and k=1, the double generalized Petersen graph is shown in Figure 1.Figure 1
The graph ofDGP13,1.


## 4. Edge Metric Dimension ofDGPn,1

In this section, we investigate the edge metric dimension ofDGPn,1 by proving its upper and lower bounds.


### 4.1. Upper Bound

For a setW=α1,α2,…,αk⊆VG, the code of any edge e∈EG is the k-vector:(1)cWe=de,α1,de,α2,…,de,αk.Equivalently, the setW is an edge resolving set for G if and only if each edge in G has the unique code with respect to W. That is, W is an edge metric for G if and only if two edges e1,e2∈EG and cWe1=cWe2 imply e1=e2. Next, first four lemmas provide the upper bound for the edge metric dimension of DGPn,1.Lemma 1.
Forn≥16 and n≡0mod4, we have βeDGPn,1≤4.Proof.
LetW=x1,xn/2−2,vn/2,yn/2−2. Define k1=n/2 and l=n/4. Codes of all the edges of DGPn,1 with respect to W are given in Tables 1–10.
From Tables1–10, it can be seen that each edge of DGPn,1 has the unique code w. r. t. W. So, W is an edge resolving set for DGPn,1 and βeDGPn,1≤4.Table 1
Codes of outermost edges,e=xcxc+1.
cWex1xn/2−2vn/2yn/2−21≤c≤k1−4c−1k1−c−3k1−ck1−c−1c=k1−3c−1033c=k1−2c−1023k1−1≤c≤k1c−1c+2−k12c+4−k1k1+1≤c≤n−3n−cc+2−k1c+1−kc+4−k1n−2≤c≤n−1n−cn−3+k1−cc+1−k1n−c+k1−1c=nn−cn−3+k1−ck1n−c+k1−1Table 2
Codes of edges,e=xcuc, when c is odd.
cWex1xn/2−2vn/2yn/2−21≤c≤k1/2−12c−2k1−2c−1k1+1−2ck1−2ck1/2≤c≤k1/2+12c−22c+1−k112c+2−k1k1/2+2≤c≤k1−1n+2−2c2c+1−k12c−k1−12c+2−k1i=k12k1−1k1−1k1Table 3
Codes of edges,e=xcuc, when c is even.
cWex1xn/2−2vn/2yn/2−21≤c≤k1/2−22c−1k1−2c−2k1+1−2ck1−2c−1k1/2−1≤c≤k1/22c−12c+2−k33k1/2+1≤c≤k1−1n+1−2c2c+2−k12c+1−k12c+3−k1c=k11k1−2k1+1k1−1Table 4
Codes of edges,e=u2c−1v2c.
cWex1xn/2−2vn/2yn/2−21≤c≤l−22c−1k1−2c−1k1−2ck1−2c−1l−1≤c≤l2c−12k1−2cc−l+2l+1≤c≤k1−1n+2−2c2c+2−k12c−k1−12c+2−k1c=k12k1−1k1−1k1−1Table 5
Codes of edges,e=u2cv2c+1.
cWex1xn/2−2vn/2yn/2−21≤c≤l−22ck1−2c−2k1+1−2ck1−2c−2l−1≤c≤l2c2c+3−k13c−l+3l+1≤c≤k1−2n+1−2c2c−k1+32c−k1+22c+3−k1c=k1−1k1+2−cn+k1−2c−2c+1n+k1−2c−2c=k1k1+2−cn+k1−2c−2c+1n+k1−2c−2Table 6
Codes of edges,e=u2cv2c−1.
cWex1xn/2−2vn/2yn/2−2c=12k1−2c−1k1+2−2ck1−2c−12≤c≤l−22c−1k1−2c−1k1+2−2ck1−2c−1l−1≤c≤l2c−1c+2−ll+3−c2l+1≤c≤k1−1n+2−2c2c+2−k12c+1−k12c+2−k1c=k12k1−1k1+1k1−1u1vn1k1−2k1−1k1−2Table 7
Codes of edges,e=u2c+1v2c.
cWex1xn/2−2vn/2yn/2−21≤c≤l−22ck1−2c−2k1−1−2ck1−2c−2l−1≤c≤l2cc+3−ll−c2c+3−k1l+1≤c≤k1−2n+1−2c2c+3−k12c−k12c+3−k1c=k1−12k1−1k1+1k1−1Table 8
Codes of edges,e=v2c−1y2c−1.
cWex1xn/2−2vn/2yn/2−2c=13k1−2k1k1−32≤c≤k1/2−12c−1k1−2ck1+2−2ck1−2c−1k1/2≤c≤k1/2+12c−12c+2−k22c+1−k1k1/2+2≤c≤k1−1n+3−2c2c+2−k12c−k12c+1−k1c=k13k1k1k1−1Table 9
Codes of edges,e=v2cy2c.
cWex1xn/2−2vn/2yn/2−21≤c≤k1/2−22ck1−2c−1k1−2ck1−2c−2k1/2−1≤c≤k1/22c3k1−2c2c+2−k1k1/2+1≤c≤k1−1n+2−2c2c+3−k12c−k12c+2−k1c=k12k1−1k1k1−2Table 10
Codes of edges,e=ycyc+1.
cWex1xn/2−2vn/2yn/2−2c=13k1−2k1−1k1−42≤c≤k1−4c+1k1−c−1k1−ck1−c−3k1−3≤c≤k1−2c+13k1−c0k1−1≤c≤k1c+1c+4−k11c+2−k1k1+1≤c≤n−3n+2−cc+4−k1c+1−k1c+2−k1n−2≤c≤n−1n+2−cn−c+k−1c+1−k1k1+n−c−3c=n3k1−1k1k1−3Lemma 2.
Forn≥13 and n≡1mod4, we have βeDGPn,1≤4.Proof.
LetW=x1,x⌊n/2⌋−1,v⌊n/2⌋+1,y⌊n/2⌋−1. Define k1=⌊n/2⌋ and l=⌊n/4⌋. Codes of all the edges of DGPn,1 with respect to W are given in Tables 11–20.
From Tables11–20, it can be observed that each edge of DGPn,1 has unique codes w. r. to W. So, W is an edge resolving set for DGPn,1 and βeDGPn,1≤4.Table 11
Codes of outermost edges,e=xcxc+1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤k1−3c−1k1−c+2k1+1−ck1−cc=k1−2c−1033k1−1≤c≤k1c−1c+1−k123c=k1+1k1224k1+2≤c≤n−2n−cc+1−k1c−k1c+3−k1n−1≤c≤nn−cn−2+k1−cc−k1n−c+k1Table 12
Codes of edges,e=xcuc, when c is odd.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤k1/2−12c−2k1−2ck1+3−2ck1+1−2ck1/2≤c≤k1/2+12c−22c−k133k1/2+2≤c≤k1n+2−2c2c−k12c−k1−12c+1−k1c=k1+11k1−1k1+1k1Table 13
Codes of edges,e=xcuc, when c is even.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤k1/2−12c−1k1−2c−1k1+1−2ck1−2cc=k1/22c−1112k1/2+1≤c≤k1−1n+1−2c2c−k1+12c−k1−12c+2−k1c=k12k1k1−1k1+1Table 14
Codes of edges,e=vcyc, when c is odd.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=13k1+1−2ck1+2−2ck1−2c2≤c≤k1/2−12c−1k1+1−2ck1+2−2ck1−2ck1/2≤c≤k1/2+12c−13k1+2−2c2c−k1k1/2+2≤c≤k1n+3−2c2c+1−k12c−k1−22c−k1c=k1+12k1k1k1−1Table 15
Codes of edges,e=vcyc, when c is even.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤k1/2−22ck1−2ck1+2−2ck1−2c−1k1/2−1≤c≤k1/22c2k1+2−2c1k1/2+1≤c≤k1−1n+2−2c2c+2−k12c−k12c+1−k1c=k13k1+1k1k1Table 16
Codes of edges,e=ycyc+1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=13k1−1k1k1−32≤c≤k1−3c+1k1−ik1+1−ck1−c−2k1−2≤c≤k1−1c+13k1+1−c0k1≤c≤k1+1c+1c+3−k11c+1−k1k1+2≤c≤n−2n+2−cc+3−k1c−k1c−k1+1n−1≤c≤n3n−c+k1c+1+k1−nk1+n−c−2Table 17
Codes of edges,e=u2c−1v2c.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−22c−1k1−2ck1+3−2ck1−2cl−1≤c≤l2c−12c−k1+13c−l+3l+1≤c≤k1n+2−2c2c−k1+12c−k12c+1−k1unv12k1−1k1k1−1Table 18
Codes of edges,e=u2cv2c+1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−32ck1−2c−1k1−2ck1−2c−1l−2≤c≤l−12c2k1−2cc−l+3l≤c≤k1−1n+1−2c2c−k1+22c−k1−12c+2−k1c=k12k1k1−1k1Table 19
Codes of edges,e=u2cv2c−1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−3c+1k1−2ck1−2c+1k1−2cl−2≤c≤l−12c−12k1−2c+1l−cl≤c≤k1n+2−2c2c+k1+2−n2c+k1−n−12c+2+k1−nu1vn1k1−1k1k1−1Table 20
Codes of edges,e=u2c+1v2c.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−32ck−2c−1k1+2−2ck1−2c−1l−2≤c≤l−12cc−2l+2−c2l≤c≤k1−1n+1−2c2c+2−k12c+1−k12c+2−k1c=k12k1k1+1k1Lemma 3.
Forn≥15 and n≡3mod4, we have βeDGPn,1≤4.Proof.
LetW=x1,x⌊n/2⌋−1,v⌊n/2⌋+1,y⌊n/2⌋−1. Define k1=⌊n/2⌋ and l=⌈n/2⌉. Codes of all the edges in DGPn,1 with respect to W are given in Tables 21–29.
From Tables21–29, we observe that each edge of DGPn,1 has unique code w. r. t. W. Hence, W is an edge resolving set for DGPn,1 and βeDGPn,1≤4.Table 21
Codes of edges,e=xcuc, when c is odd.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤⌊k1/2⌋2c−2k1−2ck1+2−2ck1+1−2cc=⌊k1/2⌋+1k1−1112⌊k1/2⌋+2≤c≤k1n+2−2c2c−k12c−k1−22c+1−k1c=k1+11k1−1k1k1Table 22
Codes of edges,e=xcuc, when c is even.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤⌊k1/2⌋−12c−1k1−2c−1k1+2−2ck1−2c⌊k1/2⌋≤c≤⌊k1/2⌋+12c−12c+1−k133⌊k1/2⌋+2≤c≤k1−1n+1−2c2c−k1+12c−k12c+2−k1c=k12k1k1k1+1Table 23
Codes of edges,e=vcyc, when c is odd.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=13k1+1−2ck1+3−2ck1−2c2≤c≤l−12c−1k1+1−2ck1+3−2ck1−2cl≤c≤l+1l+c−12c−k1+122c−k1l+2≤c≤k1n+3−2c2c+1−k12c−k1−12c−k1c=k12k1k1+1k1−1Table 24
Codes of edges,e=vcyc, when c is even.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−22ck1−2ck1+1−2ck1−2c−1l−1≤c≤l2c3k1+1−2c2c+1−k1l+1≤c≤k1−1n+2−2c2c+2−k12c−k1−12c+1−k1c=k13k1+1k1−1k1Table 25
Codes of edges,e=ycyc+1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=13k1−1k1k1−22≤c≤k1−3c+1k1−ck1+1−ck1−c−2k1−2≤c≤k1−1c+13k1+1−c0k1≤c≤k1+1c+1c+3−k11c+1−k1k1+2≤c≤n−2n+2−cc+3−k1c−k1c−k1+1n−1≤c≤n3n−c+k1c+1+k1−nk1+n−c−2Table 26
Codes of edges,e=u2c−1v2c.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−22c−1k1−2ck1+1−2ck1−2cl−1≤c≤l2c−12k1+1−2cc−l−2l+1≤c≤k1n+2−2c2c+2k1−n2c+k1−n−12c+2+k1−nunv1:2k1−1k1k1−1Table 27
Codes of edges,e=u2cv2c+1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−22ck1−2c−1k1+2−2ck1−2c−1l−1≤c≤l2c2c+3+k1−n3c−l+3l+1≤c≤k1−1n+1−2c2c−k1+22c−k1+12c+2−k1c=k1n+1−2ck1k1+1k1Table 28
Codes of edges,e=u2cv2c−1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=l2k1−2k1+1k1−22≤c≤l−22c−1k1−2ck1+3−2ck1−2cl−1≤c≤l2c−1c+2−l1+3−c2l+1≤c≤k1n+2−2c2c+2+k1−n2c+1+k1−n2c+2+k1−nu1vn:1k1−1k1k1−1Table 29
Codes of edges,e=u2c+1v2c.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=l2k1−2k1+1k1−22≤c≤l−22c−1k1−2ck1+3−2ck1−2cl−1≤c≤l2c−1c+2−l1+3−c2l+1≤c≤k1n+2−2c2c+2+k1−n2c+1+k1−n2c+2+k1−nu1vn:1k1−1k1k1−1Lemma 4.
Forn≥14 and n≡2mod4, we have βeDGPn,1≤3.Proof.
LetW=x1,xn/2+2,un/2−3. Define k1=⌊n/2⌋ and l=⌈n/4⌉. Codes of outermost edges, inner edges, and spokes of DGPn,1 with respect to W are given in Tables 30–39.
From Tables30–39, we observe that each edge of DGPn,1 has unique code w. r. t. W. Thus, W is an edge resolving set for DGPn,1 and βeDGPn,1≤3.Table 30
Codes of outermost edges,e=xcxc+1.
cWex1xn/2+2un/2−3c=1c−1k1−1k1−2c−22≤c≤k1−4c−1k1+1−ck1−c−3k1−3≤c≤k1c−1k1+1−cc+4−k1k1+1≤c≤k1+2n−c0c+4−k1k1+3≤c≤n−4n−cc+k1−n−2c+4−k1n−3≤c≤nn−ck1+c−n−2n+k1−c−3Table 31
Codes of edges,e=v2c−1y2c−1.
cWex1xn/2+2un/2−3c=13k1k1−2c−22≤c≤l−22c−1k1+4−2ck1−2−2cc=l−12c−1k1+4−2ck1−2cl≤c≤l+1k132c+2−k1l+2≤c≤k1−1n+3−2c2c−k1−22c+2−k1c=k1n+3−2c2c−k1−2k1−2Table 32
Codes of edges,e=v2cy2c.
cWex1xn/2+2un/2−31≤c≤l−32ck1−2c+3k1−2c−2c=l−22ck1+3−2c3l−1≤c≤l2ck1+3−2c2c+4−k1l+1≤c≤k1−2n+2−2c2c−k1−12c+4−k1k1−1≤c≤k1n+2−2c2c−k1−1n−2c+k1−2Table 33
Codes of spokes,e=xcuc, when c is odd.
CWex1xn/2+2un/2−3c=12c−2k1−1k1−1−2c2≤c≤l−22c−2k1+3−2ck1−2c−1l−1≤c≤l2c−2k1+3−2c2c+3−k1l+1≤c≤k1−1n+2−2c2c−k1−32c−k1+3c=k1n+2−2c2c−k1−3k1−1Table 34
Codes of spokes,e=xcuc, when c is even.
cWex1xn/2+2un/2−31≤c≤l−22c−1k1+2−2ck1−3−2cl−1≤c≤l2c−1k1+2−2c2c+3−k1l+1≤c≤k1−2n+1−2c2c−k1−22c+3−k1k1−1≤c≤k1n+1−2c2c−k1−2n+k1−2c−3Table 35
Codes of edges,e=ycyc+1.
cWex1xn/2+2un/2−3c=13k1+1k1−2c−22≤c≤k1−5c+1k1+3−ck1−c−3k1−4≤c≤k1−2c+1k1+3−c2k1−1≤c≤k1c+1k1+3−cc+4−k1k1+1≤c≤k1+2n+2−c3c+4−k1k1+3≤c≤n−4n+2−cc−k1c+4−k1n−3≤c≤n−1n+2−cc+k1−nn+4−cyny13k1k1−3Table 36
Codes of edges,e=u2c−1v2c.
cWex1xn/2+2un/2−31≤c≤l−32ck1+2−2ck1−2c−2c=l−22ck1+2−2c3l−1≤c≤ll+c−1k1+2−2c2c+5−k1l+1≤c≤k1−1n+1−2c2c−k1−12c+5−k1c=k13k1−3k1Table 37
Codes of edges,e=u2c−1v2c.
cWex1xn/2+2un/2−3c=11k1k1+1−4c2≤c≤l−3forn≠142c−1k1+3−2ck1−2c−1l−2≤c≤l−1forn=142c−1k1+3−2c3l−2≤c≤l−12c−1k1+3−2c3c=l2c−1k1+3−2c5l+1≤c≤k1−2n+2−2c2c−k1−22c+4−k1k1−1≤c≤k1n+2−2c2c−k1−2n+k1−1−2cTable 38
Codes of edges,e=u2cv2c+1.
cWex1xn/2+2un/2−31≤c≤l−32ck1+2−2ck1−4−2cl−2≤c≤l−12ck1+2−2c2c+3−k1c=l2c−124l+1≤c≤k1−2n+1−2c2c−k1−12c+3−k1c=k1−1n+2−k1−c2c−k1−1n+k1−2c−4unv1n+2−k1−c2c−k1−1n+k1−2c−4Table 39
Codes of edges,e=u2cv2c−1.
cWex1xn/2+2un/2−3c=12ck1k1−2c−32≤c≤l−22c−1k1+3−2ck1−2c−3l−1≤c≤l2c−1k1+3−2c2c+2−k1c=l+1k1−125l+2≤c≤k1−2forn≠14n+2−2c2c−k1−22c+2−k1l+2≤c≤k1forn=14n+2−2c2c−k1−2n+k1−3−2ck1−1≤c≤k1n+2−2c2c−k1−2n+k1−3−2cu1vn1k1−1k1−2


### 4.2. Lower Bound

For allt-regular graphs, the following lower bound for the edge metric dimension of any connected graph was explored in [21].Lemma 5 (see [21]).
IfG is a connected t-regular graph, then βeG≥1+⌈log2t⌉.
As we know thatDGPn,k are 3-regular graphs and ⌈log23⌉=2, so the next result holds consequently.Lemma 6.
For any double generalized Petersen graphDGPn,k, we have βeDGPn,k≥3.
The next result provides the lower bound for the edge metric dimension ofDGPn,1 whenever n≠2mod4.Lemma 7.
Forn≥13 and n≡0,1,3mod4, we have βeDGPn,1≥4.Proof.
We now prove that the cardinality of any minimum edge resolving setW is 4. On the contrary, suppose that the cardinality of W is 3, by Lemma 5. To prove that the existence of such edge resolving set W is not possible, we have the following claims. Define k=n/2, and let the vertex set of DGPn,1 be X∪Y∪U∪V, where X=xi;1≤i≤n, Y=yi;1≤i≤n, U=ui;1≤i≤n, and V=vi;1≤i≤n.Claim 1.
The setW can contain at most one vertex from either Xor Y. On contrary, let x1,xi∈W, then with out loss of generality, the third vertex of W is either from X or Y or U or V . Thus, we have following possibilities:(i)
Ifxj∈W, when n≡0,1,3mod4, then cWun−1vn=cWunvn−1 when 2≤i≤n−3,3≤j≤n−2. For n−3≤i≤n−1,n−1≤j≤n, we have the following edges with same codes: cWx1x2=cWx1u1(ii)
Ifyj∈W, when n≡0,1,3mod4, then cWun−1vn=cWunvn−1 when 2≤i≤n−2,1≤j≤n−2. For n−1≤i≤n,1≤j≤n−4, the edges with same code: cWun−3vn−2=cWun−2vn−3. Also when n−1≤i≤n,n−3≤j≤n, we have cWu2v3=cWu3v2.(iii)
Ifu2j−1∈W, when n≡0mod4 and n≥16, then cWxn−1xn=cWxnun when 2≤i≤k−1,1≤j≤k/2. For 2≤i≤k−1,k/2+1≤j≤k, cWxk−1uk−1=cWxk−1xk. When i=k,1≤j≤k/2, cWxkuk=cWxkxk+1. cWx1u1=cWxnx1 for i=k,j=k/2+1. For k≤i≤k+1,k/2+2≤j≤k, we have cWv1y1=cWy1y2. For i=k+1,j=1 and j=k/2+1, we have cWx1x2=cWxnx1. When k+1≤i≤n−1,2≤j≤k/2, cWv1y1=cWy1yn. For k+2≤i≤n,k/2+2≤j≤k and j=1, we have cWxk+2uk+2=cWxk+1xk+2. When i=k+2,j=k/2+1, the edges with same codes: cWuk+1vk=cWuk+1vk+2. Also cWxk+3uk+3=cWxk+2xk+3 when k+3≤i≤n,j=k/2+1.(iv)
Ifu2j∈W, when n≡0mod4 and n≥16, then cWxnx1=cWx1u1 when 2≤i≤k,1≤j≤k/2. For 2≤i≤k−1,k/2+1≤j≤k, cWxk+1uk+1=cWuk+1vk. When i=k,k/2+1≤j≤k, cWxkuk=cWxk+1xk. If i=k+1,j=1, then cWykyk+1=cWvk+1yk+1. For i=k+1,2≤j≤k/2, the edges with same codes: cWunv1=cWv2y2. For k+2≤i≤n,1≤j≤k/2 and k+2≤i≤n,k/2+1≤j≤k, the edges having same codes: cWxk+1xk+2=cWxk+2uk+2 and cWx1x2=cWx1u1 respectively.(v)
Ifv2j−1∈W, when n≡0mod4 and n≥16, then cWvk+1yk+1=cWykyk+1 for 2≤i≤k−1,i=k+1,1≤j≤k/2. When 2≤i≤k−2,k/2+1≤j≤k, cWvkyk=cWyk−1yk. If 3≤i≤k+1,2≤j≤k/2+1, then cWv1y1=cWy1y2. For k+3≤i≤n,k/2+2≤j≤k, the following edges have same codes: cWvk+1yk+1=cWyk+1yk+2. If k+2≤i≤n,2≤j≤k/2, then cWxk+2uk+2=cWxk+1xk+2. When k+2≤i≤n−1,k/2+1≤j≤k, then cWv1y1=cWyny1.(vi)
Ifv2j∈W, when n≡0mod4 and n≥16, then cWun−1vn=cWvnyn when 2≤i≤k−1,1≤j≤k/2−1 and j=k. If 2≤i≤k−1,1≤j≤k/2, then cWuk+1vk=cWvkyk. For 2≤i≤k−1,k/2≤j≤k−1, we have cWvn−1yn−1=cWyn−1yn. If 2≤i≤k,1≤j≤k/2−1, then cWxkuk=cWxkxk+1. When i=k,j=k/2, the edges have same codes are: cWuk−2vk−1=cxk+2xk+3. If 3≤i≤k+1,k/2+1≤j≤k, then cWv1y1=cWy1y2. When i=k+1 and k+3≤i≤n,1≤j≤k/2, then cWvk+1yk+1=cWyk+1yk+2. For k+1≤i≤n−1,1≤j≤k/2, cWv1y1=cWy1yn. If i=k+2,j=k/2+1, then cWxk−1xk=cWuk+4vk+3. For k+2≤i≤n,k/2+2≤j≤k and k+3≤i≤n,k/2+1≤j≤k, the edges with same codes: cWxk+1xk+2=cWxk+2uk+2 and cWuk+1vk+2=cWvk+2yk+2, respectively.Claim 2.
The setW can contain at most one vertex either from U or V. Due to symmetry, it is enough to show that W contains at most one vertex from U. On contrary, let u1,uj∈W.(i)
Ifuj∈W, when n≡0,1,3mod4, then cWxn−1xn=cWyn−1yn for 2≤i≤n−3,3≤j≤n−2. If n−2≤i≤n−1,n−1≤j≤n, then cWx2x3=cWy2y3.(ii)
Ifu1,u2i−1,yj∈W and , when n≡0mod4 and n≥16, then cWukvk+1=cWvk+1yk+1 when 1≤i≤k/2+1,1≤j≤k−1. If 1≤i≤k/2+1,3≤j≤k+1, then cWu2v1=cWv1y1. If 1≤i≤k/2,k+2≤j≤n, then cWxkuk=cWxkxk+1. When k/2+1≤i≤k,k+1≤j≤n−1, then cWunv1=cWy1v1. For i=1,k/2+1≤i≤k,j=1 and k+3≤j≤ncWuk+2vk+1=cWvk+1yk+1.If k/2+2≤i≤k,2≤j≤k, then cWxk+2uk+2=cWxk+1xk+2.(iii)
Ifu1,u2i,yj∈W and ,when n≡0mod4 and n≥16, then cWxk+1xk+2=cWxk+1uk+1 when 1≤i≤k/2,j=1 and k+3≤j≤. If 1≤i≤k/2,2≤j≤k, then cWxk+1uk+2=cWxk+2uk+2. When 1≤i≤k/2+1,4≤j≤k+2, then cWv2y2=cWu3v2. If k/2+1≤i≤k,1≤j≤k−1, then edges having same codes: cWxkxk+1=cWxk+1uk+1. For k/2+1≤i≤k,3≤j≤k+1 and k/2+1≤i≤k,k+2≤j≤n, then edges with same codes: cWu2v1=cWv1y1 and cWxkxk+1=cxkuk, respectively.(iv)
Ifvj∈W, when n≡0,1,3mod4, then cWxn−1xn=cWynyn−1 for 2≤i≤n−2,1≤j≤n−2. If n−1≤i≤n,1≤j≤n−4, then cWxn−3xn−2=cWyn−2yn−3. For n−2≤i≤n,n−3≤j≤n, we have cWx2x3=cWy2y3.(v)
Ifu1,u2i−1,xj∈W and when n≡0mod4 with n≥16, then cWxn−1xn=cWxnun when 2≤i≤k/2,1≤j≤k−1,j=n. If 2≤i≤k/2,1≤j≤k, then cWxkuk=cWxkxk+1. For 2≤i≤k/2,k+1≤j≤n−1, we have cWx1u1=cWu1vn. If k/2+1≤i≤k,2≤j≤k+1, then edges with same codes: cWunv1=cWv1y1. For 2≤i≤k/2+1,k+2≤j≤n and k/2+2≤i≤k,j=1 and k+2≤j≤n, then we have cWu2v1=cWv1y1 and cWxk+1xk+2=cWxk+2uk+2.(vi)
Ifu1,u2i,xj∈W, when n≡0mod4 and n≥16, then cWuk+2vk+3=cWxkxk+1 for i=1,j=1 and j=k+2. If 1≤i≤k/2,2≤j≤k+1, then cWxk+1uk+1=cWxk+1xk+2. If 1≤i≤k/2,k+2≤j≤n, then the following edges have same codes: cWxk+2uk+2=cWxk+1xk+2. For 2≤i≤k/2+1,1≤j≤2,k+4≤j≤n, then cWx4u4=cWu4v3. If k/2+1≤i≤k,1≤j≤k, then cWxkuk=cWxkxk+1. When k/2+1≤i≤k,k+1≤j≤n, then cWxk+1uk+1=cWxk+1xk.Claim 3.
Ifx1,u2i−1,yj∈W, when n≡0mod4 and n≥16, then cWx1x2=cWx1xn for i=1,1≤j≤2. If i=1 and k/2+2≤i≤k,2≤j≤k, then cWyk+2vk+2=cWuk+1vk+2. If i=1 and k/2+3≤i≤k,k/2≤j≤k+1, then edges with same codes are: cWxk+2xk+3=cWuk+4vk+3. When 1≤i≤k/2 and k+2≤j≤n, then cWvkyk=cWuk+1vk. For 2≤i≤k/2,1≤j≤k, we have cWy1v1=cWyny1. If 1≤i≤k/2,k≤j≤n−2 and j=n, then cWxn−1xn=cWxnun. Moreover, for k/2+1≤i≤k,k+1≤j≤n−1 and j=1, cWxnx1=cWx1u1, for 2≤i≤k/2+1,2≤j≤k, cWxk+1uk+1=cWuk+1vk+2 and for k/2+1≤i≤k,k+2≤j≤n, cWxk+1uk+1=cWvkuk+1.(i)
Ifx1,u2i,yj∈W, when n≡0mod4 and n≥16, then cWxnx1=cWx1u1 for 1≤i≤k/2,k+1≤j≤n−1 and j=1. If i=1,j=2, the cWu2v1=cWu2v3. For 1≤i≤k/2+1,3≤j≤k+1, we have following edges having same codes: cWxk+2uk+2=cWuk+2vk+3. If 1≤i≤k/2−1,k+2≤j≤n, then cWxk+1uk+1=cWuk+1vk. If 2≤i≤k/2+1,4≤j≤k+2 and j=2, then cWx2x3=cWx2u2. When k/2≤i≤k−1,k≤j≤n−2 and j=n, then cWxn−1xn=cWxnun. For k/2+1≤i≤k−1,1≤j≤k, cWv1y1=cWyny1. If k/2+1≤i≤k,k+2≤j≤n and j=k, then cWxkuk=cWxkxk+1. For k/2+1≤i≤k,3≤j≤k+1 and j=1, cWx1x2=cWx1u1 and for k/2+2≤i≤k,2≤j≤k, cWxk+1uk+1=cWuk+1vk+2.(ii)
Ifu1,v2i−1,yj∈W, when n≡0mod4 and n≥16, then cWu1vn=cWu1v2 for i=1 and j=1. If i=1 and k/2+1≤i≤k,2≤j≤k, then cWx1u1=cWu1vn. For i=1 and k/2s+2≤i≤k,3≤j≤k+1, the edges with same codes are: cWu2v1=cWv1y1. For 1≤i≤k/2+1 and k+2≤j≤n, cWx1u1=cWu1v2. When 2≤i≤k/2+1,k+3≤j≤n and j=1, then cWu2v3=cWv3y3. For 2≤i≤k/2+2,1≤j≤2 and k+4≤j≤, we have cWx3u3=cWu3v4. If 2≤i≤k/2,3≤j≤k, then cWxk+1xk+2=cWuk+2vk+3. For 1≤i≤k/2,k+1≤j≤n−1, we have the following edges with same codes: cWunv1=cWy1v1. If i=k/2+1,j=k+1, then cWuk+1vk=cWvk+2uk+1. For k/2+2≤i≤k,k+2≤j≤n, cWvkyk=cWuk+1vk and for k/2+1≤i≤k,1≤j≤k−1, cWukvk+1=cWyk+1vk+1.(iii)
Ifu1,v2i,yj∈W, when n≡0mod4 and n≥16, then If 1≤i≤k/2−1 and i=k,1≤j≤k−2 and j=n, then cWxn−1un−1=cWun−1vn−2. For 1≤i≤k/2−1 and i=k,k≤j≤n−2, we have cWun−1vn=cWvnyn. When 1≤i≤k/2,k+2≤j≤n, then cWvkyk=cWuk+1vk. If 1≤i≤k/2,2≤j≤k, then cWx1u1=cWu1vn. The edges cWxkxk+1=cWxk+1xk+2 have same codes when k/2≤i≤k/2+1,j=1. For 1≤i≤k/2 and i=,k+1≤j≤n−1, we have cWxkuk=cWukvk−1. If k/2+1≤i≤k,2≤j≤k, then cWvk+2yk+2=cWuk+1vk+2. For i=k/2+1,j=k+1, we have cWxk−1xk=cWuk+4vk+3. When k/2+1≤i≤k,k+2≤j≤n, then cWx1u1=cWu1v2. For k/2+2≤i≤k and i=1,1≤j≤2 and k+4≤j≤n, cWxk+3uk+3=cWuk+3vk+2 and for k/2+2≤i≤k and i=1,4≤j≤k+2, cWu3v2=cWy2v2.(iv)
Ifx1,u2i−1,v2j−1∈W, when n≡0mod4 and n≥16, then cWx2u2=cWxnun when i=j=1. If k/2+3≤i≤k,i=1,2≤j≤k/2, then cWuk+1vk+2=cWyk+2yk+3. For 1≤i≤k/2,k/2+1≤j≤k−1, we have cWun−1vn=cWvnyn. The following edges cWxkuk=cWxkxk+1 and cWun−2vn−1=cWvn−1yn−1 have same codes for 1≤i≤k/2, k/2+2≤j≤k and when i=j=k, respectively.(v)
Ifx1,u2i,v2j∈W, when n≡0mod4 and n≥16, then cWu2v1=cWu2v3 for i=j=1. If k/2+2≤i≤k and i=1,2≤j≤k/2, then cWxk+2xk+3=cWxk+3uk+3. When 1≤i≤k/2+1,k/2+1≤j≤k, then cWyk+2vk+2=cWuk+1vk+2. For 1≤i≤k/2−1 and i=k,k/2s+1≤j≤k/2+3 and for 2≤i≤k/2+2,k/2+2≤j≤k, these are the edges cWxk−1xk=cWxk−1uk−1 and cWuk+3vk+4=cWvk+4yk+4.(vi)
Ifx1,u2i−1,v2j∈W, when n≡0mod4 and n≥16, then cWykvk=cWuk+1vk when 1≤i≤k/2, and 1≤j≤k/2. If k/2+2≤i≤k and i=1,k/2+1≤j≤k, then cWyk+2vk+2=cWvk+2uk+1. When 2≤i≤k/2+1,2≤j≤k/2+1, then cWx3u3=cWu3v2. For k/2≤i≤k−1,k/2−1≤j≤k−2, cWun−3vn−2=cWxn−3un−3 and for k/2+2≤i≤k and i=1,k/2+2≤j≤k and j=1, cWu3v2=cWv2y2.(vii)
Ifx1,u2i,v2j−1∈W, when n≡0mod4 and n≥16, then cWyk−1vk−1=cWukvk−1 for 1≤i≤k/2−1 and i=k,1≤j≤k/2. If 1≤i≤k/2+1,3≤j≤k/2+1, then cWu3v2=cWv2y2. When k/2+2≤i≤k and i=1,k/2+2≤j≤k and j=1, then we have cWuk+2vk+3=cWvk+3yk+3. For i=k and 1≤i≤k/2−1,1≤j≤k/2, cWyk−1vk−1=cWvk−1uk and when k/2≤i≤k−1,k/2≤j≤k−1, then cWxn−2un−2=cWvn−1un−2.
Now, whenn≡1,3mod4, the remaining cases are as follows:Claim 4.
The setW can contain at most one vertex from either Xor Y. On contrary, let x1,xi∈W, then with out loss of generality, the third vertex of W is either from X or Y or U or V . Thus, we have following possibilities: cWxn−1xn=cWxnun for 2≤i≤k,1≤j≤k. If 2≤i≤k,j=k+1whenn≡1mod4, then cWxk+1xk+2=cWxk+1uk+1. If 2≤i≤k,j=k+1 when n≡3mod4, then cWxkxk+1=cWxkuk. For i=k+1,j=1,j=k+1, then edges with same codes are: cWy1y2=cWyn−1yn. When i=k+1,2≤j≤k, then we have cWu2v1=cWun−1vn. If k+3≤i≤n,j=1 when n≡1mod4, then cWxk+2xk+3=cWxk+3uk+3 and if k+2≤i≤n,j=1 when n≡3mod4, then cWxk+1xk+2=cWxk+2uk+2. For k+2≤i≤n,2≤j≤k+1, edges having same codes are: cWx1x2=cWx1u1.(i)
Ifu2j∈W, when n≡1,3mod4 and n≥13, then If 2≤i≤k+1,1≤j≤k, then edges with same codes are: cWxnx1=cWx1u1. For i=k+2,j=1, cWyny1=cWy2y3. The edges u3v2 and unv1 have same codes when i=k+2,2≤j≤k. For k+3≤i≤n,j=1 when n≡1mod4, cWxk+1xk+2=cWxk+2uk+2 and for k+3≤i≤n,j=1whenn≡3mod4, cWxk+2xk+3=cWxk+3uk+3. Moreover, if k+3≤i≤n,2≤j≤k, the edges having same codes are x2x3 and x2u2.(ii)
Ifv2j−1∈W, when n≡1,3mod4 and n≥13, then cWun−2vn−1=cWvn−1yn−1 for 2≤i≤k−1,1≤j≤k−1. If 2≤i≤k−1,j=k, then cWxnx1=cWx1u1. When 2≤i≤k−2,j=k+1, then edges having same codes are vn−2yn−2 and yn−2yn−1. If k−1≤i≤k+1,j=k+1, then we have cWy1v1=cWy1y2. For k+2≤i≤n,1≤j≤2 when n≡1mod4, then we have following edges with same codes cWxk+1xk+2=cWxk+2uk+2 and for k+2≤i≤n,1≤j≤2whenn≡3mod4, then ukvk+1 and uk+3vk+2 are the edges having same codes. If i=k+2,3≤j≤k, then cWunv1=cWu3v2. For i=k+2,j=k+1 and for k+3≤i≤n,3≤j≤k+1, it can be seen that, cWukvk+1=cWuk+1vk and cWu3v2=cWv2y2, respectively.(iii)
Ifv2j∈W, when n≡1,3mod4 and n≥13, then cWun−1vn=cWvnyn for 2≤i≤k,1≤j≤k−1. If 2≤i≤k−1 and i=n,j=k, then we have cWun−2vn−1=cWyn−1vn−1. When 2≤i≤k+1,j=k, then the edges cWxk+1xk+2=cWxk+1uk+1 have same codes with respect to W. If i=k+1,2≤j≤k−1, cWu2v1=cWun−1vn. When i=k+1,j=1 and n≡1mod4, we have cWukvk+1=cuk+3vk+2. When i=k+1,j=1 and n≡3mod4, then cWxk+1xk+2=cWxk+1uk+1. If k+2≤i≤n,2≤j≤k, then cWx1u1=cWx1x2. If k+2≤i≤n−1,1≤j≤k and i=n,1≤j≤k−1, then cWv1y1=cWyny1 and cWxnun=cWxn−1xn.Claim 5.
The setW contains at most one vertex either from Uor V . Due to symmetry, it is enough to show that W contains at most one vertex from U. On the contrary, let u1,uj∈W.(i)
Ifu1,u2i−1,yj∈W and , when n≡1,3mod4 and n≥13, then cWunvn−1=cWvn−1yn−1 for 2≤i≤k,1≤j≤2. If i=k+1,1≤j≤2 and when n≡1mod4, then cWxk−1xk=cWuk+1vk. For i=k+1,1≤j≤2 and when n≡3mod4, then we have cWxk−1xk=cWukvk+1. If 2≤i≤k+1,3≤j≤k+1, then cWu2v1=cWv1y1. When 2≤i≤k+1,k+1≤j≤n−2, then edges having same codes are: cWun−1vn=cWvnyn. For 2≤i≤k−1,n−1≤j≤n, we have cWun−2vn−3=cWvn−3yn−3. If k≤i≤k+1,n−1≤j≤n when n≡1mod4 and if k≤i≤k+1,n−1≤j≤n when n≡3mod4, then we have cWykyk+1=cWuk+2vk+1 and cWykyk+1=cWuk+1vk+2, respectively.(ii)
Ifu1,u2i,yj∈W and, when n≡1,3mod4 and n≥13, then For 1≤i≤k,4≤j≤k+1, we have cWu3v2=cWv2y2. If 1≤i≤k,k+2≤j≤n−1, then cWunv1=cWy1v1. When 2≤i≤k,j=n, then cWx1u1=cWu1v2. If 2≤i≤k,j=1, cWu2v3=cWy3v3. For i=1,1≤j≤2 and j=n when n≡1mod4, then we have cWxk+3xk+4=cWuk+2vk+3. If i=1,1≤j≤2 and j=n when n≡3mod4, then cWxk+3xk+4=cWuk+3vk+2. When 1≤i≤k−4,2≤j≤3 when n≡1mod4, then cWuk+1vk+2=cWyk+2yk+3. For 1≤i≤k−4,2≤j≤3 when n≡3mod4, then cWuk+2vk+1=cWyk+2yk+3. Moreover, if k−3≤i≤k,2≤j≤3, then cWu4v5=cWv5y5.(iii)
Ifu1,u2i−1,xj∈W and , when n≡1,3mod4 and n≥13, then for 2≤i≤k,1≤j≤k and j=n, the edges having same codes are: cWxnun=cWxn−1xn. If 2≤i≤k,k+1≤j≤n−1, then cWxn−1xn=cWxn−1un−1. If i=k+1,1≤j≤k, then cWvnyn=cWun−1vn. When i=k+1,k+2≤j≤n and i=k+1,j=k+1, the edges having same codes are: cWv1y1=cWu2v1 and cWx2u2=cWxn−1un−1, respectively.(iv)
Ifu1,u2i,xj∈W, when n≡1,3mod4 and n≥13, then cWunv1=cWv1y1 when i=1,2≤j≤k+1. If i=1,j=k+2, then cWxnun=cWx3u3. For i=1,k+3≤j≤n and j=1, then cWu3v2=cWv2y2. If 2≤i≤k,3≤j≤k+2 and 2≤i≤k,1≤j≤2 and k+3≤j≤n, then we have cWx2x3=cWx3u3 and cWx2x3=cWx2u2.Claim 6.
Ifx1,u2i−1,yj∈W, when n≡1,3mod4 and n≥13, then cWxn−1xn=cWxnun for 1≤i≤k,k+1≤j≤n−2. When k−1≤i≤k+1,k+1≤j≤n when n≡1mod4, then cWykyk+1=cWyk+1vk+1. If k−2≤i≤k+1,k≤j≤n−1 when n≡3mod4, then cWyk−1yk=cWykvk. For 2≤i≤k,1≤j≤k+1, we have cWv1y1=cWyny1. If i=1,3≤j≤k+1 when n≡1mod4, then cWxk+2xk+3=cWuk+3vk+4 and when i=1,3≤j≤k+1 when n≡3mod4, then cWxk+2xk+3=cWuk+4vk+3. When n≡1mod4 and 1≤i≤k−3,n−1≤j≤n, the edges are cWxk+1xk+2=cWukvk+1. When n≡3mod4 and 1≤i≤k−3,n−1≤j≤n, the edges with same codes are: cWyk+1vk+1=cWuk+2vk+1. If n≡1mod4 and i=k−2,n−1≤j≤n, then cWxk+1uk+1=cWuk+1vk. When n≡3mod4 and k−2≤i≤k+1,n−1≤j≤n, then cWukvk+1=cWxk+1xk+2. For i=1,1≤j≤2 and i=1,j=3, cWunv1=cWu2v1 and cWxk+2xk+3=cWuk+3vk+4.(i)
Ifx1,u2i,yj∈W, when n≡1,3mod4 and n≥13, then if 1≤i≤k,k+2≤j≤n−1, then cWxnx1=cWx1u1. When 1≤i≤k−1,k≤j≤k+1, then cWvn−1yn−1=cWyn−1yn. For 3≤i≤k,4≤j≤k+1, we have cWy3y4=cWy4v4. If 1≤i≤k−2,2≤j≤k−1 when n≡1mod4, then we have cWyk+2vk+2=cWuk+1vk+2. When n≡3mod4 and 1≤i≤k−2,2≤j≤k−1, then cWyk+3vk+3=cWuk+2vk+3. If k−1≤i≤k,3≤j≤k−1, then cWuk+2vk+3=cWyk+3vk+3. For 2≤i≤k,j=1 and j=n, cWy1v1=cy1y2. When 2≤i≤k,j=2, then cWx2u2=cWx2x3. For i=1,j=n when n≡1mod4 and i=1,j=n when n≡3mod4, cWyk+1vk+1=cWuk+2vk+1 and cWukvk+1=cWxk+1xk+2.(ii)
Ifu1,v2i−1,yj∈W, when n≡1,3mod4 and n≥13, then If 1≤i≤k+1,k+2≤j≤n, then cWx1u1=cWu1v2. If 3≤i≤k+1,4≤j≤k+1, then cWu3v2=cWv2y2. For 3≤i≤k+1,1≤j≤3, we have cWv3y3=cWy3y4. When n≡1mod4 and 1≤i≤2,1≤j≤k, then cWxk+1xk+2=cWxk+2uk+2 and when 1≤i≤3, 1≤j≤k−1 and n≡3mod4, then cWuk+1vk+2=cWxkxk+1. If i=1,k≤j≤k+1 and n≡3mod4, then the edges with same codes are: cWx1u1=cu1vn and if i=2,k≤j≤k+1 and n≡3mod4, then cWxk+2xk+3=cWuk+4vk+3. For n≡1mod4 and i=1,j=k−2 and when n≡1mod4 and i=2,j=k+1, the edges having same codes are: cWx1u1=cWu1vn and cWxk+2xk+3=cWuk+3vk+4.(iii)
Ifu1,v2i,yj∈W, when n≡1,3mod4 and n≥13, then for 1≤i≤k,2≤j≤k+1, cWx1u1=cWu1vn. If 1≤i≤k−1,k+2≤j≤n−2, then cWun−1vn=cWvnyn. When i=k,k+2≤j≤n−1, then cWukvk−1=cWxkxk+1. For 1≤i≤k−1,j=1 and n−1≤j≤n, the edges with same codes are: cWvn−1yn−1=cWyn−2yn−1. If i=k,j=1,n, then cWxn−1un−1=cWun−1vn−2.(iv)
Ifx1,u2i−1,v2j∈W, when n≡1,3mod4 and n≥13, then cWu3v2=cWv2y2 if 1≤i≤k3≤j≤k+1. For i=1,j=k, cWxn−2xn−1=cWv1u2. For 2≤i≤k+1,j=k, cWx1u1=cWx1x2.(v)
Ifx1,u2i,v2j−1∈W, when n≡1,3mod4 and n≥13, then If 1≤i≤k,3≤j≤k+1, then we have cWu3v2=cWv2y2. For 1≤i≤k,1≤j≤2, cWun−2vn−1=cWvn−1yn−1.(vi)
Ifx1,u2i−1,v2j−1∈W, when n≡1,3mod4 and n≥13, then cWvn−4yn−4=cWun−5vn−4 for 1≤i≤k−1,k−1≤j≤k+1. if n≡1mod4 and k−2≤i≤k+1,1≤j≤k−2, then cWvk+1yk+1=cWvk+1uk+2. When n≡3mod4 and k−2≤i≤k+1,1≤j≤k−3, then cWvkyk=cWvkuk+1. When n≡1mod4 and i=k+1,k−1≤i≤k+1, then cWxk+1xk+2=cWukvk+1. If n≡3mod4 and i=k+1,k−1≤i≤k+1, then cWxk+1xk+2=cWuk+1vk.(vii)
Ifx1,u2i,v2j∈W and n≡1,3mod4 with n≥13: for n≡1mod4 and 1≤i≤k−2,k−2≤j≤k, we have cWvk+3yk+3=cWvk+3uk+2. Whenever 1≤i≤k−2,k−2≤j≤k, and cWvk+1yk+1=cWvk+1uk+2. Whenever k−3≤i≤k,j≤k−3. For n≡1,3mod4 and i=k,k−2≤j≤k, we have cWxkxk+1=cWuk−1vkcWxkxk+1=cWukvk−1, cWvk+3yk+3=cWvk+3uk+2. If k−3≤i≤k,1≤j≤k−3, then cWvk+1yk+1=cWvk+1uk+2.
Forn≡1mod4 and i=k,k−2≤i≤k and for n≡3mod4 and i=k,k−2≤i, cWxkxk+1=cWuk−1vkcWxkxk+1=cWukvk−1.
From all these claims, it yields that there is no such edge resolving setW for DGPn,1. Therefore βeDGPn,1≥4.
From Lemmas1–6, we have the following main result:Theorem 1.
Forn≥13, we have (2)βeDGPn,1=3,whenn≡2mod4,4,otherwise.


## 4.1. Upper Bound

For a setW=α1,α2,…,αk⊆VG, the code of any edge e∈EG is the k-vector:(1)cWe=de,α1,de,α2,…,de,αk.Equivalently, the setW is an edge resolving set for G if and only if each edge in G has the unique code with respect to W. That is, W is an edge metric for G if and only if two edges e1,e2∈EG and cWe1=cWe2 imply e1=e2. Next, first four lemmas provide the upper bound for the edge metric dimension of DGPn,1.Lemma 1.
Forn≥16 and n≡0mod4, we have βeDGPn,1≤4.Proof.
LetW=x1,xn/2−2,vn/2,yn/2−2. Define k1=n/2 and l=n/4. Codes of all the edges of DGPn,1 with respect to W are given in Tables 1–10.
From Tables1–10, it can be seen that each edge of DGPn,1 has the unique code w. r. t. W. So, W is an edge resolving set for DGPn,1 and βeDGPn,1≤4.Table 1
Codes of outermost edges,e=xcxc+1.
cWex1xn/2−2vn/2yn/2−21≤c≤k1−4c−1k1−c−3k1−ck1−c−1c=k1−3c−1033c=k1−2c−1023k1−1≤c≤k1c−1c+2−k12c+4−k1k1+1≤c≤n−3n−cc+2−k1c+1−kc+4−k1n−2≤c≤n−1n−cn−3+k1−cc+1−k1n−c+k1−1c=nn−cn−3+k1−ck1n−c+k1−1Table 2
Codes of edges,e=xcuc, when c is odd.
cWex1xn/2−2vn/2yn/2−21≤c≤k1/2−12c−2k1−2c−1k1+1−2ck1−2ck1/2≤c≤k1/2+12c−22c+1−k112c+2−k1k1/2+2≤c≤k1−1n+2−2c2c+1−k12c−k1−12c+2−k1i=k12k1−1k1−1k1Table 3
Codes of edges,e=xcuc, when c is even.
cWex1xn/2−2vn/2yn/2−21≤c≤k1/2−22c−1k1−2c−2k1+1−2ck1−2c−1k1/2−1≤c≤k1/22c−12c+2−k33k1/2+1≤c≤k1−1n+1−2c2c+2−k12c+1−k12c+3−k1c=k11k1−2k1+1k1−1Table 4
Codes of edges,e=u2c−1v2c.
cWex1xn/2−2vn/2yn/2−21≤c≤l−22c−1k1−2c−1k1−2ck1−2c−1l−1≤c≤l2c−12k1−2cc−l+2l+1≤c≤k1−1n+2−2c2c+2−k12c−k1−12c+2−k1c=k12k1−1k1−1k1−1Table 5
Codes of edges,e=u2cv2c+1.
cWex1xn/2−2vn/2yn/2−21≤c≤l−22ck1−2c−2k1+1−2ck1−2c−2l−1≤c≤l2c2c+3−k13c−l+3l+1≤c≤k1−2n+1−2c2c−k1+32c−k1+22c+3−k1c=k1−1k1+2−cn+k1−2c−2c+1n+k1−2c−2c=k1k1+2−cn+k1−2c−2c+1n+k1−2c−2Table 6
Codes of edges,e=u2cv2c−1.
cWex1xn/2−2vn/2yn/2−2c=12k1−2c−1k1+2−2ck1−2c−12≤c≤l−22c−1k1−2c−1k1+2−2ck1−2c−1l−1≤c≤l2c−1c+2−ll+3−c2l+1≤c≤k1−1n+2−2c2c+2−k12c+1−k12c+2−k1c=k12k1−1k1+1k1−1u1vn1k1−2k1−1k1−2Table 7
Codes of edges,e=u2c+1v2c.
cWex1xn/2−2vn/2yn/2−21≤c≤l−22ck1−2c−2k1−1−2ck1−2c−2l−1≤c≤l2cc+3−ll−c2c+3−k1l+1≤c≤k1−2n+1−2c2c+3−k12c−k12c+3−k1c=k1−12k1−1k1+1k1−1Table 8
Codes of edges,e=v2c−1y2c−1.
cWex1xn/2−2vn/2yn/2−2c=13k1−2k1k1−32≤c≤k1/2−12c−1k1−2ck1+2−2ck1−2c−1k1/2≤c≤k1/2+12c−12c+2−k22c+1−k1k1/2+2≤c≤k1−1n+3−2c2c+2−k12c−k12c+1−k1c=k13k1k1k1−1Table 9
Codes of edges,e=v2cy2c.
cWex1xn/2−2vn/2yn/2−21≤c≤k1/2−22ck1−2c−1k1−2ck1−2c−2k1/2−1≤c≤k1/22c3k1−2c2c+2−k1k1/2+1≤c≤k1−1n+2−2c2c+3−k12c−k12c+2−k1c=k12k1−1k1k1−2Table 10
Codes of edges,e=ycyc+1.
cWex1xn/2−2vn/2yn/2−2c=13k1−2k1−1k1−42≤c≤k1−4c+1k1−c−1k1−ck1−c−3k1−3≤c≤k1−2c+13k1−c0k1−1≤c≤k1c+1c+4−k11c+2−k1k1+1≤c≤n−3n+2−cc+4−k1c+1−k1c+2−k1n−2≤c≤n−1n+2−cn−c+k−1c+1−k1k1+n−c−3c=n3k1−1k1k1−3Lemma 2.
Forn≥13 and n≡1mod4, we have βeDGPn,1≤4.Proof.
LetW=x1,x⌊n/2⌋−1,v⌊n/2⌋+1,y⌊n/2⌋−1. Define k1=⌊n/2⌋ and l=⌊n/4⌋. Codes of all the edges of DGPn,1 with respect to W are given in Tables 11–20.
From Tables11–20, it can be observed that each edge of DGPn,1 has unique codes w. r. to W. So, W is an edge resolving set for DGPn,1 and βeDGPn,1≤4.Table 11
Codes of outermost edges,e=xcxc+1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤k1−3c−1k1−c+2k1+1−ck1−cc=k1−2c−1033k1−1≤c≤k1c−1c+1−k123c=k1+1k1224k1+2≤c≤n−2n−cc+1−k1c−k1c+3−k1n−1≤c≤nn−cn−2+k1−cc−k1n−c+k1Table 12
Codes of edges,e=xcuc, when c is odd.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤k1/2−12c−2k1−2ck1+3−2ck1+1−2ck1/2≤c≤k1/2+12c−22c−k133k1/2+2≤c≤k1n+2−2c2c−k12c−k1−12c+1−k1c=k1+11k1−1k1+1k1Table 13
Codes of edges,e=xcuc, when c is even.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤k1/2−12c−1k1−2c−1k1+1−2ck1−2cc=k1/22c−1112k1/2+1≤c≤k1−1n+1−2c2c−k1+12c−k1−12c+2−k1c=k12k1k1−1k1+1Table 14
Codes of edges,e=vcyc, when c is odd.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=13k1+1−2ck1+2−2ck1−2c2≤c≤k1/2−12c−1k1+1−2ck1+2−2ck1−2ck1/2≤c≤k1/2+12c−13k1+2−2c2c−k1k1/2+2≤c≤k1n+3−2c2c+1−k12c−k1−22c−k1c=k1+12k1k1k1−1Table 15
Codes of edges,e=vcyc, when c is even.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤k1/2−22ck1−2ck1+2−2ck1−2c−1k1/2−1≤c≤k1/22c2k1+2−2c1k1/2+1≤c≤k1−1n+2−2c2c+2−k12c−k12c+1−k1c=k13k1+1k1k1Table 16
Codes of edges,e=ycyc+1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=13k1−1k1k1−32≤c≤k1−3c+1k1−ik1+1−ck1−c−2k1−2≤c≤k1−1c+13k1+1−c0k1≤c≤k1+1c+1c+3−k11c+1−k1k1+2≤c≤n−2n+2−cc+3−k1c−k1c−k1+1n−1≤c≤n3n−c+k1c+1+k1−nk1+n−c−2Table 17
Codes of edges,e=u2c−1v2c.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−22c−1k1−2ck1+3−2ck1−2cl−1≤c≤l2c−12c−k1+13c−l+3l+1≤c≤k1n+2−2c2c−k1+12c−k12c+1−k1unv12k1−1k1k1−1Table 18
Codes of edges,e=u2cv2c+1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−32ck1−2c−1k1−2ck1−2c−1l−2≤c≤l−12c2k1−2cc−l+3l≤c≤k1−1n+1−2c2c−k1+22c−k1−12c+2−k1c=k12k1k1−1k1Table 19
Codes of edges,e=u2cv2c−1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−3c+1k1−2ck1−2c+1k1−2cl−2≤c≤l−12c−12k1−2c+1l−cl≤c≤k1n+2−2c2c+k1+2−n2c+k1−n−12c+2+k1−nu1vn1k1−1k1k1−1Table 20
Codes of edges,e=u2c+1v2c.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−32ck−2c−1k1+2−2ck1−2c−1l−2≤c≤l−12cc−2l+2−c2l≤c≤k1−1n+1−2c2c+2−k12c+1−k12c+2−k1c=k12k1k1+1k1Lemma 3.
Forn≥15 and n≡3mod4, we have βeDGPn,1≤4.Proof.
LetW=x1,x⌊n/2⌋−1,v⌊n/2⌋+1,y⌊n/2⌋−1. Define k1=⌊n/2⌋ and l=⌈n/2⌉. Codes of all the edges in DGPn,1 with respect to W are given in Tables 21–29.
From Tables21–29, we observe that each edge of DGPn,1 has unique code w. r. t. W. Hence, W is an edge resolving set for DGPn,1 and βeDGPn,1≤4.Table 21
Codes of edges,e=xcuc, when c is odd.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤⌊k1/2⌋2c−2k1−2ck1+2−2ck1+1−2cc=⌊k1/2⌋+1k1−1112⌊k1/2⌋+2≤c≤k1n+2−2c2c−k12c−k1−22c+1−k1c=k1+11k1−1k1k1Table 22
Codes of edges,e=xcuc, when c is even.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤⌊k1/2⌋−12c−1k1−2c−1k1+2−2ck1−2c⌊k1/2⌋≤c≤⌊k1/2⌋+12c−12c+1−k133⌊k1/2⌋+2≤c≤k1−1n+1−2c2c−k1+12c−k12c+2−k1c=k12k1k1k1+1Table 23
Codes of edges,e=vcyc, when c is odd.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=13k1+1−2ck1+3−2ck1−2c2≤c≤l−12c−1k1+1−2ck1+3−2ck1−2cl≤c≤l+1l+c−12c−k1+122c−k1l+2≤c≤k1n+3−2c2c+1−k12c−k1−12c−k1c=k12k1k1+1k1−1Table 24
Codes of edges,e=vcyc, when c is even.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−22ck1−2ck1+1−2ck1−2c−1l−1≤c≤l2c3k1+1−2c2c+1−k1l+1≤c≤k1−1n+2−2c2c+2−k12c−k1−12c+1−k1c=k13k1+1k1−1k1Table 25
Codes of edges,e=ycyc+1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=13k1−1k1k1−22≤c≤k1−3c+1k1−ck1+1−ck1−c−2k1−2≤c≤k1−1c+13k1+1−c0k1≤c≤k1+1c+1c+3−k11c+1−k1k1+2≤c≤n−2n+2−cc+3−k1c−k1c−k1+1n−1≤c≤n3n−c+k1c+1+k1−nk1+n−c−2Table 26
Codes of edges,e=u2c−1v2c.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−22c−1k1−2ck1+1−2ck1−2cl−1≤c≤l2c−12k1+1−2cc−l−2l+1≤c≤k1n+2−2c2c+2k1−n2c+k1−n−12c+2+k1−nunv1:2k1−1k1k1−1Table 27
Codes of edges,e=u2cv2c+1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−11≤c≤l−22ck1−2c−1k1+2−2ck1−2c−1l−1≤c≤l2c2c+3+k1−n3c−l+3l+1≤c≤k1−1n+1−2c2c−k1+22c−k1+12c+2−k1c=k1n+1−2ck1k1+1k1Table 28
Codes of edges,e=u2cv2c−1.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=l2k1−2k1+1k1−22≤c≤l−22c−1k1−2ck1+3−2ck1−2cl−1≤c≤l2c−1c+2−l1+3−c2l+1≤c≤k1n+2−2c2c+2+k1−n2c+1+k1−n2c+2+k1−nu1vn:1k1−1k1k1−1Table 29
Codes of edges,e=u2c+1v2c.
cWex1x⌊n/2⌋−1v⌊n/2⌋+1y⌊n/2⌋−1c=l2k1−2k1+1k1−22≤c≤l−22c−1k1−2ck1+3−2ck1−2cl−1≤c≤l2c−1c+2−l1+3−c2l+1≤c≤k1n+2−2c2c+2+k1−n2c+1+k1−n2c+2+k1−nu1vn:1k1−1k1k1−1Lemma 4.
Forn≥14 and n≡2mod4, we have βeDGPn,1≤3.Proof.
LetW=x1,xn/2+2,un/2−3. Define k1=⌊n/2⌋ and l=⌈n/4⌉. Codes of outermost edges, inner edges, and spokes of DGPn,1 with respect to W are given in Tables 30–39.
From Tables30–39, we observe that each edge of DGPn,1 has unique code w. r. t. W. Thus, W is an edge resolving set for DGPn,1 and βeDGPn,1≤3.Table 30
Codes of outermost edges,e=xcxc+1.
cWex1xn/2+2un/2−3c=1c−1k1−1k1−2c−22≤c≤k1−4c−1k1+1−ck1−c−3k1−3≤c≤k1c−1k1+1−cc+4−k1k1+1≤c≤k1+2n−c0c+4−k1k1+3≤c≤n−4n−cc+k1−n−2c+4−k1n−3≤c≤nn−ck1+c−n−2n+k1−c−3Table 31
Codes of edges,e=v2c−1y2c−1.
cWex1xn/2+2un/2−3c=13k1k1−2c−22≤c≤l−22c−1k1+4−2ck1−2−2cc=l−12c−1k1+4−2ck1−2cl≤c≤l+1k132c+2−k1l+2≤c≤k1−1n+3−2c2c−k1−22c+2−k1c=k1n+3−2c2c−k1−2k1−2Table 32
Codes of edges,e=v2cy2c.
cWex1xn/2+2un/2−31≤c≤l−32ck1−2c+3k1−2c−2c=l−22ck1+3−2c3l−1≤c≤l2ck1+3−2c2c+4−k1l+1≤c≤k1−2n+2−2c2c−k1−12c+4−k1k1−1≤c≤k1n+2−2c2c−k1−1n−2c+k1−2Table 33
Codes of spokes,e=xcuc, when c is odd.
CWex1xn/2+2un/2−3c=12c−2k1−1k1−1−2c2≤c≤l−22c−2k1+3−2ck1−2c−1l−1≤c≤l2c−2k1+3−2c2c+3−k1l+1≤c≤k1−1n+2−2c2c−k1−32c−k1+3c=k1n+2−2c2c−k1−3k1−1Table 34
Codes of spokes,e=xcuc, when c is even.
cWex1xn/2+2un/2−31≤c≤l−22c−1k1+2−2ck1−3−2cl−1≤c≤l2c−1k1+2−2c2c+3−k1l+1≤c≤k1−2n+1−2c2c−k1−22c+3−k1k1−1≤c≤k1n+1−2c2c−k1−2n+k1−2c−3Table 35
Codes of edges,e=ycyc+1.
cWex1xn/2+2un/2−3c=13k1+1k1−2c−22≤c≤k1−5c+1k1+3−ck1−c−3k1−4≤c≤k1−2c+1k1+3−c2k1−1≤c≤k1c+1k1+3−cc+4−k1k1+1≤c≤k1+2n+2−c3c+4−k1k1+3≤c≤n−4n+2−cc−k1c+4−k1n−3≤c≤n−1n+2−cc+k1−nn+4−cyny13k1k1−3Table 36
Codes of edges,e=u2c−1v2c.
cWex1xn/2+2un/2−31≤c≤l−32ck1+2−2ck1−2c−2c=l−22ck1+2−2c3l−1≤c≤ll+c−1k1+2−2c2c+5−k1l+1≤c≤k1−1n+1−2c2c−k1−12c+5−k1c=k13k1−3k1Table 37
Codes of edges,e=u2c−1v2c.
cWex1xn/2+2un/2−3c=11k1k1+1−4c2≤c≤l−3forn≠142c−1k1+3−2ck1−2c−1l−2≤c≤l−1forn=142c−1k1+3−2c3l−2≤c≤l−12c−1k1+3−2c3c=l2c−1k1+3−2c5l+1≤c≤k1−2n+2−2c2c−k1−22c+4−k1k1−1≤c≤k1n+2−2c2c−k1−2n+k1−1−2cTable 38
Codes of edges,e=u2cv2c+1.
cWex1xn/2+2un/2−31≤c≤l−32ck1+2−2ck1−4−2cl−2≤c≤l−12ck1+2−2c2c+3−k1c=l2c−124l+1≤c≤k1−2n+1−2c2c−k1−12c+3−k1c=k1−1n+2−k1−c2c−k1−1n+k1−2c−4unv1n+2−k1−c2c−k1−1n+k1−2c−4Table 39
Codes of edges,e=u2cv2c−1.
cWex1xn/2+2un/2−3c=12ck1k1−2c−32≤c≤l−22c−1k1+3−2ck1−2c−3l−1≤c≤l2c−1k1+3−2c2c+2−k1c=l+1k1−125l+2≤c≤k1−2forn≠14n+2−2c2c−k1−22c+2−k1l+2≤c≤k1forn=14n+2−2c2c−k1−2n+k1−3−2ck1−1≤c≤k1n+2−2c2c−k1−2n+k1−3−2cu1vn1k1−1k1−2


## 4.2. Lower Bound

For allt-regular graphs, the following lower bound for the edge metric dimension of any connected graph was explored in [21].Lemma 5 (see [21]).
IfG is a connected t-regular graph, then βeG≥1+⌈log2t⌉.
As we know thatDGPn,k are 3-regular graphs and ⌈log23⌉=2, so the next result holds consequently.Lemma 6.
For any double generalized Petersen graphDGPn,k, we have βeDGPn,k≥3.
The next result provides the lower bound for the edge metric dimension ofDGPn,1 whenever n≠2mod4.Lemma 7.
Forn≥13 and n≡0,1,3mod4, we have βeDGPn,1≥4.Proof.
We now prove that the cardinality of any minimum edge resolving setW is 4. On the contrary, suppose that the cardinality of W is 3, by Lemma 5. To prove that the existence of such edge resolving set W is not possible, we have the following claims. Define k=n/2, and let the vertex set of DGPn,1 be X∪Y∪U∪V, where X=xi;1≤i≤n, Y=yi;1≤i≤n, U=ui;1≤i≤n, and V=vi;1≤i≤n.Claim 1.
The setW can contain at most one vertex from either Xor Y. On contrary, let x1,xi∈W, then with out loss of generality, the third vertex of W is either from X or Y or U or V . Thus, we have following possibilities:(i)
Ifxj∈W, when n≡0,1,3mod4, then cWun−1vn=cWunvn−1 when 2≤i≤n−3,3≤j≤n−2. For n−3≤i≤n−1,n−1≤j≤n, we have the following edges with same codes: cWx1x2=cWx1u1(ii)
Ifyj∈W, when n≡0,1,3mod4, then cWun−1vn=cWunvn−1 when 2≤i≤n−2,1≤j≤n−2. For n−1≤i≤n,1≤j≤n−4, the edges with same code: cWun−3vn−2=cWun−2vn−3. Also when n−1≤i≤n,n−3≤j≤n, we have cWu2v3=cWu3v2.(iii)
Ifu2j−1∈W, when n≡0mod4 and n≥16, then cWxn−1xn=cWxnun when 2≤i≤k−1,1≤j≤k/2. For 2≤i≤k−1,k/2+1≤j≤k, cWxk−1uk−1=cWxk−1xk. When i=k,1≤j≤k/2, cWxkuk=cWxkxk+1. cWx1u1=cWxnx1 for i=k,j=k/2+1. For k≤i≤k+1,k/2+2≤j≤k, we have cWv1y1=cWy1y2. For i=k+1,j=1 and j=k/2+1, we have cWx1x2=cWxnx1. When k+1≤i≤n−1,2≤j≤k/2, cWv1y1=cWy1yn. For k+2≤i≤n,k/2+2≤j≤k and j=1, we have cWxk+2uk+2=cWxk+1xk+2. When i=k+2,j=k/2+1, the edges with same codes: cWuk+1vk=cWuk+1vk+2. Also cWxk+3uk+3=cWxk+2xk+3 when k+3≤i≤n,j=k/2+1.(iv)
Ifu2j∈W, when n≡0mod4 and n≥16, then cWxnx1=cWx1u1 when 2≤i≤k,1≤j≤k/2. For 2≤i≤k−1,k/2+1≤j≤k, cWxk+1uk+1=cWuk+1vk. When i=k,k/2+1≤j≤k, cWxkuk=cWxk+1xk. If i=k+1,j=1, then cWykyk+1=cWvk+1yk+1. For i=k+1,2≤j≤k/2, the edges with same codes: cWunv1=cWv2y2. For k+2≤i≤n,1≤j≤k/2 and k+2≤i≤n,k/2+1≤j≤k, the edges having same codes: cWxk+1xk+2=cWxk+2uk+2 and cWx1x2=cWx1u1 respectively.(v)
Ifv2j−1∈W, when n≡0mod4 and n≥16, then cWvk+1yk+1=cWykyk+1 for 2≤i≤k−1,i=k+1,1≤j≤k/2. When 2≤i≤k−2,k/2+1≤j≤k, cWvkyk=cWyk−1yk. If 3≤i≤k+1,2≤j≤k/2+1, then cWv1y1=cWy1y2. For k+3≤i≤n,k/2+2≤j≤k, the following edges have same codes: cWvk+1yk+1=cWyk+1yk+2. If k+2≤i≤n,2≤j≤k/2, then cWxk+2uk+2=cWxk+1xk+2. When k+2≤i≤n−1,k/2+1≤j≤k, then cWv1y1=cWyny1.(vi)
Ifv2j∈W, when n≡0mod4 and n≥16, then cWun−1vn=cWvnyn when 2≤i≤k−1,1≤j≤k/2−1 and j=k. If 2≤i≤k−1,1≤j≤k/2, then cWuk+1vk=cWvkyk. For 2≤i≤k−1,k/2≤j≤k−1, we have cWvn−1yn−1=cWyn−1yn. If 2≤i≤k,1≤j≤k/2−1, then cWxkuk=cWxkxk+1. When i=k,j=k/2, the edges have same codes are: cWuk−2vk−1=cxk+2xk+3. If 3≤i≤k+1,k/2+1≤j≤k, then cWv1y1=cWy1y2. When i=k+1 and k+3≤i≤n,1≤j≤k/2, then cWvk+1yk+1=cWyk+1yk+2. For k+1≤i≤n−1,1≤j≤k/2, cWv1y1=cWy1yn. If i=k+2,j=k/2+1, then cWxk−1xk=cWuk+4vk+3. For k+2≤i≤n,k/2+2≤j≤k and k+3≤i≤n,k/2+1≤j≤k, the edges with same codes: cWxk+1xk+2=cWxk+2uk+2 and cWuk+1vk+2=cWvk+2yk+2, respectively.Claim 2.
The setW can contain at most one vertex either from U or V. Due to symmetry, it is enough to show that W contains at most one vertex from U. On contrary, let u1,uj∈W.(i)
Ifuj∈W, when n≡0,1,3mod4, then cWxn−1xn=cWyn−1yn for 2≤i≤n−3,3≤j≤n−2. If n−2≤i≤n−1,n−1≤j≤n, then cWx2x3=cWy2y3.(ii)
Ifu1,u2i−1,yj∈W and , when n≡0mod4 and n≥16, then cWukvk+1=cWvk+1yk+1 when 1≤i≤k/2+1,1≤j≤k−1. If 1≤i≤k/2+1,3≤j≤k+1, then cWu2v1=cWv1y1. If 1≤i≤k/2,k+2≤j≤n, then cWxkuk=cWxkxk+1. When k/2+1≤i≤k,k+1≤j≤n−1, then cWunv1=cWy1v1. For i=1,k/2+1≤i≤k,j=1 and k+3≤j≤ncWuk+2vk+1=cWvk+1yk+1.If k/2+2≤i≤k,2≤j≤k, then cWxk+2uk+2=cWxk+1xk+2.(iii)
Ifu1,u2i,yj∈W and ,when n≡0mod4 and n≥16, then cWxk+1xk+2=cWxk+1uk+1 when 1≤i≤k/2,j=1 and k+3≤j≤. If 1≤i≤k/2,2≤j≤k, then cWxk+1uk+2=cWxk+2uk+2. When 1≤i≤k/2+1,4≤j≤k+2, then cWv2y2=cWu3v2. If k/2+1≤i≤k,1≤j≤k−1, then edges having same codes: cWxkxk+1=cWxk+1uk+1. For k/2+1≤i≤k,3≤j≤k+1 and k/2+1≤i≤k,k+2≤j≤n, then edges with same codes: cWu2v1=cWv1y1 and cWxkxk+1=cxkuk, respectively.(iv)
Ifvj∈W, when n≡0,1,3mod4, then cWxn−1xn=cWynyn−1 for 2≤i≤n−2,1≤j≤n−2. If n−1≤i≤n,1≤j≤n−4, then cWxn−3xn−2=cWyn−2yn−3. For n−2≤i≤n,n−3≤j≤n, we have cWx2x3=cWy2y3.(v)
Ifu1,u2i−1,xj∈W and when n≡0mod4 with n≥16, then cWxn−1xn=cWxnun when 2≤i≤k/2,1≤j≤k−1,j=n. If 2≤i≤k/2,1≤j≤k, then cWxkuk=cWxkxk+1. For 2≤i≤k/2,k+1≤j≤n−1, we have cWx1u1=cWu1vn. If k/2+1≤i≤k,2≤j≤k+1, then edges with same codes: cWunv1=cWv1y1. For 2≤i≤k/2+1,k+2≤j≤n and k/2+2≤i≤k,j=1 and k+2≤j≤n, then we have cWu2v1=cWv1y1 and cWxk+1xk+2=cWxk+2uk+2.(vi)
Ifu1,u2i,xj∈W, when n≡0mod4 and n≥16, then cWuk+2vk+3=cWxkxk+1 for i=1,j=1 and j=k+2. If 1≤i≤k/2,2≤j≤k+1, then cWxk+1uk+1=cWxk+1xk+2. If 1≤i≤k/2,k+2≤j≤n, then the following edges have same codes: cWxk+2uk+2=cWxk+1xk+2. For 2≤i≤k/2+1,1≤j≤2,k+4≤j≤n, then cWx4u4=cWu4v3. If k/2+1≤i≤k,1≤j≤k, then cWxkuk=cWxkxk+1. When k/2+1≤i≤k,k+1≤j≤n, then cWxk+1uk+1=cWxk+1xk.Claim 3.
Ifx1,u2i−1,yj∈W, when n≡0mod4 and n≥16, then cWx1x2=cWx1xn for i=1,1≤j≤2. If i=1 and k/2+2≤i≤k,2≤j≤k, then cWyk+2vk+2=cWuk+1vk+2. If i=1 and k/2+3≤i≤k,k/2≤j≤k+1, then edges with same codes are: cWxk+2xk+3=cWuk+4vk+3. When 1≤i≤k/2 and k+2≤j≤n, then cWvkyk=cWuk+1vk. For 2≤i≤k/2,1≤j≤k, we have cWy1v1=cWyny1. If 1≤i≤k/2,k≤j≤n−2 and j=n, then cWxn−1xn=cWxnun. Moreover, for k/2+1≤i≤k,k+1≤j≤n−1 and j=1, cWxnx1=cWx1u1, for 2≤i≤k/2+1,2≤j≤k, cWxk+1uk+1=cWuk+1vk+2 and for k/2+1≤i≤k,k+2≤j≤n, cWxk+1uk+1=cWvkuk+1.(i)
Ifx1,u2i,yj∈W, when n≡0mod4 and n≥16, then cWxnx1=cWx1u1 for 1≤i≤k/2,k+1≤j≤n−1 and j=1. If i=1,j=2, the cWu2v1=cWu2v3. For 1≤i≤k/2+1,3≤j≤k+1, we have following edges having same codes: cWxk+2uk+2=cWuk+2vk+3. If 1≤i≤k/2−1,k+2≤j≤n, then cWxk+1uk+1=cWuk+1vk. If 2≤i≤k/2+1,4≤j≤k+2 and j=2, then cWx2x3=cWx2u2. When k/2≤i≤k−1,k≤j≤n−2 and j=n, then cWxn−1xn=cWxnun. For k/2+1≤i≤k−1,1≤j≤k, cWv1y1=cWyny1. If k/2+1≤i≤k,k+2≤j≤n and j=k, then cWxkuk=cWxkxk+1. For k/2+1≤i≤k,3≤j≤k+1 and j=1, cWx1x2=cWx1u1 and for k/2+2≤i≤k,2≤j≤k, cWxk+1uk+1=cWuk+1vk+2.(ii)
Ifu1,v2i−1,yj∈W, when n≡0mod4 and n≥16, then cWu1vn=cWu1v2 for i=1 and j=1. If i=1 and k/2+1≤i≤k,2≤j≤k, then cWx1u1=cWu1vn. For i=1 and k/2s+2≤i≤k,3≤j≤k+1, the edges with same codes are: cWu2v1=cWv1y1. For 1≤i≤k/2+1 and k+2≤j≤n, cWx1u1=cWu1v2. When 2≤i≤k/2+1,k+3≤j≤n and j=1, then cWu2v3=cWv3y3. For 2≤i≤k/2+2,1≤j≤2 and k+4≤j≤, we have cWx3u3=cWu3v4. If 2≤i≤k/2,3≤j≤k, then cWxk+1xk+2=cWuk+2vk+3. For 1≤i≤k/2,k+1≤j≤n−1, we have the following edges with same codes: cWunv1=cWy1v1. If i=k/2+1,j=k+1, then cWuk+1vk=cWvk+2uk+1. For k/2+2≤i≤k,k+2≤j≤n, cWvkyk=cWuk+1vk and for k/2+1≤i≤k,1≤j≤k−1, cWukvk+1=cWyk+1vk+1.(iii)
Ifu1,v2i,yj∈W, when n≡0mod4 and n≥16, then If 1≤i≤k/2−1 and i=k,1≤j≤k−2 and j=n, then cWxn−1un−1=cWun−1vn−2. For 1≤i≤k/2−1 and i=k,k≤j≤n−2, we have cWun−1vn=cWvnyn. When 1≤i≤k/2,k+2≤j≤n, then cWvkyk=cWuk+1vk. If 1≤i≤k/2,2≤j≤k, then cWx1u1=cWu1vn. The edges cWxkxk+1=cWxk+1xk+2 have same codes when k/2≤i≤k/2+1,j=1. For 1≤i≤k/2 and i=,k+1≤j≤n−1, we have cWxkuk=cWukvk−1. If k/2+1≤i≤k,2≤j≤k, then cWvk+2yk+2=cWuk+1vk+2. For i=k/2+1,j=k+1, we have cWxk−1xk=cWuk+4vk+3. When k/2+1≤i≤k,k+2≤j≤n, then cWx1u1=cWu1v2. For k/2+2≤i≤k and i=1,1≤j≤2 and k+4≤j≤n, cWxk+3uk+3=cWuk+3vk+2 and for k/2+2≤i≤k and i=1,4≤j≤k+2, cWu3v2=cWy2v2.(iv)
Ifx1,u2i−1,v2j−1∈W, when n≡0mod4 and n≥16, then cWx2u2=cWxnun when i=j=1. If k/2+3≤i≤k,i=1,2≤j≤k/2, then cWuk+1vk+2=cWyk+2yk+3. For 1≤i≤k/2,k/2+1≤j≤k−1, we have cWun−1vn=cWvnyn. The following edges cWxkuk=cWxkxk+1 and cWun−2vn−1=cWvn−1yn−1 have same codes for 1≤i≤k/2, k/2+2≤j≤k and when i=j=k, respectively.(v)
Ifx1,u2i,v2j∈W, when n≡0mod4 and n≥16, then cWu2v1=cWu2v3 for i=j=1. If k/2+2≤i≤k and i=1,2≤j≤k/2, then cWxk+2xk+3=cWxk+3uk+3. When 1≤i≤k/2+1,k/2+1≤j≤k, then cWyk+2vk+2=cWuk+1vk+2. For 1≤i≤k/2−1 and i=k,k/2s+1≤j≤k/2+3 and for 2≤i≤k/2+2,k/2+2≤j≤k, these are the edges cWxk−1xk=cWxk−1uk−1 and cWuk+3vk+4=cWvk+4yk+4.(vi)
Ifx1,u2i−1,v2j∈W, when n≡0mod4 and n≥16, then cWykvk=cWuk+1vk when 1≤i≤k/2, and 1≤j≤k/2. If k/2+2≤i≤k and i=1,k/2+1≤j≤k, then cWyk+2vk+2=cWvk+2uk+1. When 2≤i≤k/2+1,2≤j≤k/2+1, then cWx3u3=cWu3v2. For k/2≤i≤k−1,k/2−1≤j≤k−2, cWun−3vn−2=cWxn−3un−3 and for k/2+2≤i≤k and i=1,k/2+2≤j≤k and j=1, cWu3v2=cWv2y2.(vii)
Ifx1,u2i,v2j−1∈W, when n≡0mod4 and n≥16, then cWyk−1vk−1=cWukvk−1 for 1≤i≤k/2−1 and i=k,1≤j≤k/2. If 1≤i≤k/2+1,3≤j≤k/2+1, then cWu3v2=cWv2y2. When k/2+2≤i≤k and i=1,k/2+2≤j≤k and j=1, then we have cWuk+2vk+3=cWvk+3yk+3. For i=k and 1≤i≤k/2−1,1≤j≤k/2, cWyk−1vk−1=cWvk−1uk and when k/2≤i≤k−1,k/2≤j≤k−1, then cWxn−2un−2=cWvn−1un−2.
Now, whenn≡1,3mod4, the remaining cases are as follows:Claim 4.
The setW can contain at most one vertex from either Xor Y. On contrary, let x1,xi∈W, then with out loss of generality, the third vertex of W is either from X or Y or U or V . Thus, we have following possibilities: cWxn−1xn=cWxnun for 2≤i≤k,1≤j≤k. If 2≤i≤k,j=k+1whenn≡1mod4, then cWxk+1xk+2=cWxk+1uk+1. If 2≤i≤k,j=k+1 when n≡3mod4, then cWxkxk+1=cWxkuk. For i=k+1,j=1,j=k+1, then edges with same codes are: cWy1y2=cWyn−1yn. When i=k+1,2≤j≤k, then we have cWu2v1=cWun−1vn. If k+3≤i≤n,j=1 when n≡1mod4, then cWxk+2xk+3=cWxk+3uk+3 and if k+2≤i≤n,j=1 when n≡3mod4, then cWxk+1xk+2=cWxk+2uk+2. For k+2≤i≤n,2≤j≤k+1, edges having same codes are: cWx1x2=cWx1u1.(i)
Ifu2j∈W, when n≡1,3mod4 and n≥13, then If 2≤i≤k+1,1≤j≤k, then edges with same codes are: cWxnx1=cWx1u1. For i=k+2,j=1, cWyny1=cWy2y3. The edges u3v2 and unv1 have same codes when i=k+2,2≤j≤k. For k+3≤i≤n,j=1 when n≡1mod4, cWxk+1xk+2=cWxk+2uk+2 and for k+3≤i≤n,j=1whenn≡3mod4, cWxk+2xk+3=cWxk+3uk+3. Moreover, if k+3≤i≤n,2≤j≤k, the edges having same codes are x2x3 and x2u2.(ii)
Ifv2j−1∈W, when n≡1,3mod4 and n≥13, then cWun−2vn−1=cWvn−1yn−1 for 2≤i≤k−1,1≤j≤k−1. If 2≤i≤k−1,j=k, then cWxnx1=cWx1u1. When 2≤i≤k−2,j=k+1, then edges having same codes are vn−2yn−2 and yn−2yn−1. If k−1≤i≤k+1,j=k+1, then we have cWy1v1=cWy1y2. For k+2≤i≤n,1≤j≤2 when n≡1mod4, then we have following edges with same codes cWxk+1xk+2=cWxk+2uk+2 and for k+2≤i≤n,1≤j≤2whenn≡3mod4, then ukvk+1 and uk+3vk+2 are the edges having same codes. If i=k+2,3≤j≤k, then cWunv1=cWu3v2. For i=k+2,j=k+1 and for k+3≤i≤n,3≤j≤k+1, it can be seen that, cWukvk+1=cWuk+1vk and cWu3v2=cWv2y2, respectively.(iii)
Ifv2j∈W, when n≡1,3mod4 and n≥13, then cWun−1vn=cWvnyn for 2≤i≤k,1≤j≤k−1. If 2≤i≤k−1 and i=n,j=k, then we have cWun−2vn−1=cWyn−1vn−1. When 2≤i≤k+1,j=k, then the edges cWxk+1xk+2=cWxk+1uk+1 have same codes with respect to W. If i=k+1,2≤j≤k−1, cWu2v1=cWun−1vn. When i=k+1,j=1 and n≡1mod4, we have cWukvk+1=cuk+3vk+2. When i=k+1,j=1 and n≡3mod4, then cWxk+1xk+2=cWxk+1uk+1. If k+2≤i≤n,2≤j≤k, then cWx1u1=cWx1x2. If k+2≤i≤n−1,1≤j≤k and i=n,1≤j≤k−1, then cWv1y1=cWyny1 and cWxnun=cWxn−1xn.Claim 5.
The setW contains at most one vertex either from Uor V . Due to symmetry, it is enough to show that W contains at most one vertex from U. On the contrary, let u1,uj∈W.(i)
Ifu1,u2i−1,yj∈W and , when n≡1,3mod4 and n≥13, then cWunvn−1=cWvn−1yn−1 for 2≤i≤k,1≤j≤2. If i=k+1,1≤j≤2 and when n≡1mod4, then cWxk−1xk=cWuk+1vk. For i=k+1,1≤j≤2 and when n≡3mod4, then we have cWxk−1xk=cWukvk+1. If 2≤i≤k+1,3≤j≤k+1, then cWu2v1=cWv1y1. When 2≤i≤k+1,k+1≤j≤n−2, then edges having same codes are: cWun−1vn=cWvnyn. For 2≤i≤k−1,n−1≤j≤n, we have cWun−2vn−3=cWvn−3yn−3. If k≤i≤k+1,n−1≤j≤n when n≡1mod4 and if k≤i≤k+1,n−1≤j≤n when n≡3mod4, then we have cWykyk+1=cWuk+2vk+1 and cWykyk+1=cWuk+1vk+2, respectively.(ii)
Ifu1,u2i,yj∈W and, when n≡1,3mod4 and n≥13, then For 1≤i≤k,4≤j≤k+1, we have cWu3v2=cWv2y2. If 1≤i≤k,k+2≤j≤n−1, then cWunv1=cWy1v1. When 2≤i≤k,j=n, then cWx1u1=cWu1v2. If 2≤i≤k,j=1, cWu2v3=cWy3v3. For i=1,1≤j≤2 and j=n when n≡1mod4, then we have cWxk+3xk+4=cWuk+2vk+3. If i=1,1≤j≤2 and j=n when n≡3mod4, then cWxk+3xk+4=cWuk+3vk+2. When 1≤i≤k−4,2≤j≤3 when n≡1mod4, then cWuk+1vk+2=cWyk+2yk+3. For 1≤i≤k−4,2≤j≤3 when n≡3mod4, then cWuk+2vk+1=cWyk+2yk+3. Moreover, if k−3≤i≤k,2≤j≤3, then cWu4v5=cWv5y5.(iii)
Ifu1,u2i−1,xj∈W and , when n≡1,3mod4 and n≥13, then for 2≤i≤k,1≤j≤k and j=n, the edges having same codes are: cWxnun=cWxn−1xn. If 2≤i≤k,k+1≤j≤n−1, then cWxn−1xn=cWxn−1un−1. If i=k+1,1≤j≤k, then cWvnyn=cWun−1vn. When i=k+1,k+2≤j≤n and i=k+1,j=k+1, the edges having same codes are: cWv1y1=cWu2v1 and cWx2u2=cWxn−1un−1, respectively.(iv)
Ifu1,u2i,xj∈W, when n≡1,3mod4 and n≥13, then cWunv1=cWv1y1 when i=1,2≤j≤k+1. If i=1,j=k+2, then cWxnun=cWx3u3. For i=1,k+3≤j≤n and j=1, then cWu3v2=cWv2y2. If 2≤i≤k,3≤j≤k+2 and 2≤i≤k,1≤j≤2 and k+3≤j≤n, then we have cWx2x3=cWx3u3 and cWx2x3=cWx2u2.Claim 6.
Ifx1,u2i−1,yj∈W, when n≡1,3mod4 and n≥13, then cWxn−1xn=cWxnun for 1≤i≤k,k+1≤j≤n−2. When k−1≤i≤k+1,k+1≤j≤n when n≡1mod4, then cWykyk+1=cWyk+1vk+1. If k−2≤i≤k+1,k≤j≤n−1 when n≡3mod4, then cWyk−1yk=cWykvk. For 2≤i≤k,1≤j≤k+1, we have cWv1y1=cWyny1. If i=1,3≤j≤k+1 when n≡1mod4, then cWxk+2xk+3=cWuk+3vk+4 and when i=1,3≤j≤k+1 when n≡3mod4, then cWxk+2xk+3=cWuk+4vk+3. When n≡1mod4 and 1≤i≤k−3,n−1≤j≤n, the edges are cWxk+1xk+2=cWukvk+1. When n≡3mod4 and 1≤i≤k−3,n−1≤j≤n, the edges with same codes are: cWyk+1vk+1=cWuk+2vk+1. If n≡1mod4 and i=k−2,n−1≤j≤n, then cWxk+1uk+1=cWuk+1vk. When n≡3mod4 and k−2≤i≤k+1,n−1≤j≤n, then cWukvk+1=cWxk+1xk+2. For i=1,1≤j≤2 and i=1,j=3, cWunv1=cWu2v1 and cWxk+2xk+3=cWuk+3vk+4.(i)
Ifx1,u2i,yj∈W, when n≡1,3mod4 and n≥13, then if 1≤i≤k,k+2≤j≤n−1, then cWxnx1=cWx1u1. When 1≤i≤k−1,k≤j≤k+1, then cWvn−1yn−1=cWyn−1yn. For 3≤i≤k,4≤j≤k+1, we have cWy3y4=cWy4v4. If 1≤i≤k−2,2≤j≤k−1 when n≡1mod4, then we have cWyk+2vk+2=cWuk+1vk+2. When n≡3mod4 and 1≤i≤k−2,2≤j≤k−1, then cWyk+3vk+3=cWuk+2vk+3. If k−1≤i≤k,3≤j≤k−1, then cWuk+2vk+3=cWyk+3vk+3. For 2≤i≤k,j=1 and j=n, cWy1v1=cy1y2. When 2≤i≤k,j=2, then cWx2u2=cWx2x3. For i=1,j=n when n≡1mod4 and i=1,j=n when n≡3mod4, cWyk+1vk+1=cWuk+2vk+1 and cWukvk+1=cWxk+1xk+2.(ii)
Ifu1,v2i−1,yj∈W, when n≡1,3mod4 and n≥13, then If 1≤i≤k+1,k+2≤j≤n, then cWx1u1=cWu1v2. If 3≤i≤k+1,4≤j≤k+1, then cWu3v2=cWv2y2. For 3≤i≤k+1,1≤j≤3, we have cWv3y3=cWy3y4. When n≡1mod4 and 1≤i≤2,1≤j≤k, then cWxk+1xk+2=cWxk+2uk+2 and when 1≤i≤3, 1≤j≤k−1 and n≡3mod4, then cWuk+1vk+2=cWxkxk+1. If i=1,k≤j≤k+1 and n≡3mod4, then the edges with same codes are: cWx1u1=cu1vn and if i=2,k≤j≤k+1 and n≡3mod4, then cWxk+2xk+3=cWuk+4vk+3. For n≡1mod4 and i=1,j=k−2 and when n≡1mod4 and i=2,j=k+1, the edges having same codes are: cWx1u1=cWu1vn and cWxk+2xk+3=cWuk+3vk+4.(iii)
Ifu1,v2i,yj∈W, when n≡1,3mod4 and n≥13, then for 1≤i≤k,2≤j≤k+1, cWx1u1=cWu1vn. If 1≤i≤k−1,k+2≤j≤n−2, then cWun−1vn=cWvnyn. When i=k,k+2≤j≤n−1, then cWukvk−1=cWxkxk+1. For 1≤i≤k−1,j=1 and n−1≤j≤n, the edges with same codes are: cWvn−1yn−1=cWyn−2yn−1. If i=k,j=1,n, then cWxn−1un−1=cWun−1vn−2.(iv)
Ifx1,u2i−1,v2j∈W, when n≡1,3mod4 and n≥13, then cWu3v2=cWv2y2 if 1≤i≤k3≤j≤k+1. For i=1,j=k, cWxn−2xn−1=cWv1u2. For 2≤i≤k+1,j=k, cWx1u1=cWx1x2.(v)
Ifx1,u2i,v2j−1∈W, when n≡1,3mod4 and n≥13, then If 1≤i≤k,3≤j≤k+1, then we have cWu3v2=cWv2y2. For 1≤i≤k,1≤j≤2, cWun−2vn−1=cWvn−1yn−1.(vi)
Ifx1,u2i−1,v2j−1∈W, when n≡1,3mod4 and n≥13, then cWvn−4yn−4=cWun−5vn−4 for 1≤i≤k−1,k−1≤j≤k+1. if n≡1mod4 and k−2≤i≤k+1,1≤j≤k−2, then cWvk+1yk+1=cWvk+1uk+2. When n≡3mod4 and k−2≤i≤k+1,1≤j≤k−3, then cWvkyk=cWvkuk+1. When n≡1mod4 and i=k+1,k−1≤i≤k+1, then cWxk+1xk+2=cWukvk+1. If n≡3mod4 and i=k+1,k−1≤i≤k+1, then cWxk+1xk+2=cWuk+1vk.(vii)
Ifx1,u2i,v2j∈W and n≡1,3mod4 with n≥13: for n≡1mod4 and 1≤i≤k−2,k−2≤j≤k, we have cWvk+3yk+3=cWvk+3uk+2. Whenever 1≤i≤k−2,k−2≤j≤k, and cWvk+1yk+1=cWvk+1uk+2. Whenever k−3≤i≤k,j≤k−3. For n≡1,3mod4 and i=k,k−2≤j≤k, we have cWxkxk+1=cWuk−1vkcWxkxk+1=cWukvk−1, cWvk+3yk+3=cWvk+3uk+2. If k−3≤i≤k,1≤j≤k−3, then cWvk+1yk+1=cWvk+1uk+2.
Forn≡1mod4 and i=k,k−2≤i≤k and for n≡3mod4 and i=k,k−2≤i, cWxkxk+1=cWuk−1vkcWxkxk+1=cWukvk−1.
From all these claims, it yields that there is no such edge resolving setW for DGPn,1. Therefore βeDGPn,1≥4.
From Lemmas1–6, we have the following main result:Theorem 1.
Forn≥13, we have (2)βeDGPn,1=3,whenn≡2mod4,4,otherwise.


## 5. Conclusion

In this study, a family of double generalized Petersen graphDGPn,1 has been considered in the context of edge resolvability. It has been investigated that minimum 4 vertices perform the edge resolvability in DGPn,1 when n≡0,1,2 (mod 4) and minimum 3 vertices perform the edge resolvability in DGPn,1 when n≡2 (mod 4).



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*Source: 6490698-2022-04-22.xml*