# TheK-Size Edge Metric Dimension of Graphs

Authors: Tanveer Iqbal; Muhammad Naeem Azhar; Syed Ahtsham Ul Haq Bokhary
Journal: Journal of Mathematics
 (2020)

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## Abstract

In this paper, a new conceptk-size edge resolving set for a connected graph G in the context of resolvability of graphs is defined. Some properties and realizable results on k-size edge resolvability of graphs are studied. The existence of this new parameter in different graphs is investigated, and the k-size edge metric dimension of path, cycle, and complete bipartite graph is computed. It is shown that these families have unbounded k-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 k-size edge metric dimension.


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## Body

## 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 G is the minimum cardinality of edge resolving set, say X, and is denoted as βeG. An edge metric generator for G of cardinality βeG is an edge metric basis for G [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 x is identified by a unique edge e in a metric basis X, but if at some point the communication between the vertex x and edge e is blocked, then the vertex x cannot be accessed by the edge metric basis X. 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 k-size resolving sets in graphs by Naeem et al. [2], we present a new concept in the context of edge resolvabililty, called the k-size edge resolving set in graphs.For an undirected, simple, and connected graphG, the vertex set is VG and edge set is EG. The distance parameter in graphs has been used to distinguish (resolve or determine) the vertices or edges of G. The distance between the vertex α and the edge β=α1α2 in a graph G is given by dGβ,α=mindGα1,α,dGα2,α. Any two edges β1 and β2 are resolved by a vertex α of a graph G, whenever dGα,β1≠dGα,β2. A set of vertices X is an edge metric generator for a graph G, whenever every two edges of G are resolved by some vertex of X. The edge metric dimension of graph G is the minimum cardinality of set X and is denoted as βeG. An edge metric generator for G of cardinality βeG is an edge metric basis for G [1].Definition 1.
A set of verticesW is said to be a k-size edge resolving set of a graph G of order n≥2 if W is an edge resolving set and the size of subgraph W induced by W is equal to k. The k-size edge metric dimension of G, denoted by βkseG, is the minimum cardinality of a k-size edge resolving set of G. Moreover, the k-size edge resolving set of cardinality βkseG is represented as kser-set, where k≥1 and belongs to a set of natural number.
Now, we discuss the existence of this new parameter in some simple, nontrivial connected graphs.
LetH be a connected graph having the vertex set VH=l1,l2,…,lp,s1,s2,…,sq∪t1,t2 and edge set EH=l1l2,lit1,pjt1,pjt2:1≤i≤p,1≤j≤q, as shown in Figure 1. A set W=l1,l2,…,lp,s2,s3,…,sq is a 1ser-set for H. It can be seen that, for 3≤p≤q, the k-size edge resolving set for graph H exist only for k=1.
We observe that the setS=q1,q2,q3 is a minimum 2ser-set for G of Figure 1. Moreover, for the graph G, 1ser-set does not exist.
We have the following remark from these two examples.Figure 1
The graphsG and H.Remark 1.
(i)
The existence ofβkseG does not imply the existence of βk+tseG for t≥1 and vice versa in any nontrivial connected graph G(ii)
2≤βkseG≤n, for any simple graph G, where 1≤k≤n2
In this paper, we compute thek-size edge metric dimension in several well known families of graphs, Cartesian product graphs Pm□Pn,Pm□Cn, and generalized Petersen graphs GPq,1. Moreover, we present some realizable result on k-size edge metric dimension in graphs for k=1,2.


## 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 ofK-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 theirk-size edge metric dimension.Lemma 1.
For a path graphG=Pn1≤k≤n−1, βkseG=k+1ifn≥2.Proof.
Consider a path graphPn with vertex set VPn=l1,l2,…,ln and edge set EPn=lili+1:1≤i≤n−1. Let W=l1,l2,l3,…,lk+1 be a subset of vertex set of Pn. The code of each edge lsls+11≤s≤k with respect to W is distinct because each edge has the 0 entry at its sth and s+1th place. The code of each edge lsls+1 is cWlsls+1=l−1,l−2,l−3,…,l−k+1, for k+2≤s≤n−1, and cWlsls+1=l−1,l−2,l−3,…,2,1,0, for s=k+1. Thus, W is an edge resolving set for Pn. Since the size of Pn is n−1, it implies that the subgraph W induced by W has k edges. Therefore, W is k-size edge resolving set for Pn. Hence, βksePn=k+1 for 1≤k≤n−1.Lemma 2.
For a simple and connected graphG of order n≥6, βkseG=k+1ifG=Cn1≤k≤n−2.Proof.
LetCn:t1,t2,t3,…,tn,t1 be a cycle graph of order n≥6, and let W=t1,t2,t3,…,tk+1. We define l=n/2. The code of each edge tptp+1 for 1≤p≤k1≤k≤l and for 1≤p≤ll+1≤k≤n−2 with respect to W has the 0 entry at its pth and p+1th place. The code of each edge tptp+1 is cWtptp+1=p−1,p−2,p−3,…,p−k+1, for k+1≤p≤l1≤k≤l−1,. The qth and q+1th entries of edges tptp+1 are equal to l−1 when n is even 1≤q≤k,p=l+q,1≤k≤l,1≤q≤l,l+1≤k≤n−2 and equal to l−2 and l−1, respectively, when n is odd 1≤q≤k,p=l+q,q,1≤k≤l,1≤q≤l−1,l+1≤k≤n−2. The code of remaining edges tptp+1 is n−p,n+1−p,n+2−p,…,n+k−p, for l+k+1≤p≤n,1≤k≤l−1, when n is even and, for 1≤k≤l−2, when n is odd. We note that the codes of all the edges are distinct. Therefore, W is a k-size edge resolving set for Cn. Hence, βkseCn=k+1 for 1≤k≤n−2.
Thek-size edge resolving sets of a complete bipartite graph Ks1,s2 exist only for the values of k given in the following result.Lemma 3.
For the complete bipartite graphKs1,s2s1=s2 and s1,s2≥4,(1)βkseKs1,s2=2s1−2,k=s1−1s2−1,2s1−1,k=s1s2−s1ork=s1s2−1,2s1,k=s1s2,while, for s1≥s2≥1, we have(2)βkseKs1,s2=s1+k1−1,k=k1s1−1,1≤k1≤s1−1,s1+k2,k=k2s1,1≤k2≤s1−1.Observation. It cannot be necessary that a k-size edge resolving set has at least k vertices in it.To justify our above observation, we consider a graphG=K4. A set of vertices W=f1,f2,f3,f4 is the vertex set of G=K4. One can observe that the set VK4 is a 6-size edge resolving set for K4. Therefore, β6seK4=4.


## 4.K-Size Edge Metric Dimension of Cartesian Product of Graphs

LetG=Pm□Pn be the Cartesian product of two path graphs Pm and Pn, for m,n≥4. Let E1G=rghrgh+1:1≤g≤m,1≤h≤n be the set of horizontal edges and E2G=sghsg+1h:1≤g≤m−1,1≤h≤n be the set of vertical edges of Pm□Pn. The graph of P5□P6 is shown in Figure 2. To find distances, we embed G into xy plane in such a way that each vertex is in an ordered pair form. Let the vertices 0,0,m−1,0,0,n−1,m−1,n−1 be the corner vertices of G. In the next two lemmas, we shall discuss size 1, size 2, and size 3 edge metric dimension of G=Pm□Pn and G=Pm□Cn, for m,n≥5.Figure 2
G=P5□P6.Lemma 4.
LetG be the cartesian product graph Pm□Pn; then, we have(3)βksePm□Pn=3,k=1,4,k=2,5,k=3.Proof.
Here, we will prove this result fork=3 (only). For this we consider W=a1,a2,a3,a4,a5, where a1=0,0,a2=1,0,a3=m−1,0,a4=0,1,anda5=0,2, and prove that W is a 3ser-set for Pm□Pn. Note that dr1,s1,r2,s2=r1−r2+s1−s2 is the distance between any two vertices of Pm□Pn. Let α=r1,s1r2,s2 be an edge. The distances of the edge α from the vertices of W are calculated as follows.
da1,α=r1+s1,da3,α=m−1−r2+s1, da2,α=r1+s1−10,1 when α∈rghrgh+1,sghsg+1h:1≤g≤m,2≤h≤n, da2,α=r1+s1 when α∈rghrgh+1:1≤g≤m,h=1, and da2,α=r1+s1+1 when α∈sghsg+1h:1≤g≤m−1,h=1; da4,α=r1+s1+1, whenever α∈rghrgh+1:g=1,1≤h≤n, da4,α=r1+s1−1, whenever α∈rghrgh+1,sghsg+1h:2≤g≤m,1≤h≤n, da4,α=r1+s1, whenever α∈sghsg+1h:g=1,1≤h≤n, da5,α=r1+s1+2e, whenever α∈rghrgh+1:g=1,1≤h≤n, da5,α=r1+s1, whenever α∈rghrgh+1:g=2,1≤h≤n, da5,α=r1+s1+1, whenever α∈sghsg+1h:g=1,1≤h≤n, da5,α=r1+s1−1, whenever α∈sghsg+1h:g=2,1≤h≤n, and da5,α=r1+s1−2, whenever α∈rghrgh+1,sghsg+1h:3≤g≤m,1≤h≤n. Suppose contrary two edges α1=r1,s1r2,s2 and α2=t1,u1t2,u2 are at the same distance from the vertices of W. Thus, we have the following equalities:(4)r1+s1=t1+u1,m−1−r2+s1=m−1−t2+u1.
The above equalities imply thats1−u1=r2−t2. Thus, it follows that r1+r2=t1+t2. In both the cases r1=r2 or r1=r2−1, and we get r1=t1 and r2=t2. The equality r1=t1 together with r1+s1=t1+u1 implies that s1=u1. Both the vertices s2 and u2 can either equal to s1 or equal s1+1. One of the edges α1 or α2 does not represent an edge if they have distinct values. So, finally we have α1=α2, which is a contradiction. Therefore, W is an edge resolving set for Pm□Pn. Moreover, the subgraph induced by W has 3 edges. Hence, we conclude that β3sePm□Pn=5. Similarly, we can prove the result for the values of k=1,2. Hence, we conclude the result.
We present the following result on thek-size edge metric dimension of the Cartesian product graph Pm□Cn without proof.Lemma 5.
LetG be the cartesian product graph Pm□Cn; then, we have(5)βksePm□Cn=3,k=1,2,4,k=3.


## 5.K-Size Edge Metric Dimension of Generalized Petersen Graphs

The generalized Petersen graphGPq,s is a 3-regular graph containing 2q vertices and 3q edges. The vertex set of GPq,s is VGPq,s=ct,dt:1≤t≤q, and the edge set is EGPq,s=ctct+1,ctdt,dtdt+s:1≤t≤q,1≤s<q/2. The edges ctdt for 1≤t≤q are said to be spokes of GPq,s. The outer cycle of GPq,s is said to be the principal cycle GPq,s.We will compute thek-size edge metric dimension of GPq,s for k=1,2 and s=1 in the following two sections. Firstly, we will find upper bound for the 1-size edge metric dimension of GPq,1.Lemma 6.
For allq≥7, we have β1seGPq,1≤3.Proof.
LetW=c1,c3,d1 be a set of vertices of GPq,1.Case 1.
Whenq is odd.
Here, we definet1=q+1/2. There is only one edge in the subgraph W induced by W and the codes of all the edges of GPq,1 are given in Tables 1–3.Table 1
Codes of outer edges ofGPq,1 when q is odd.
d.,.c1c3d1c1c2011ctct+1:2≤t≤3t−10tcttt+1:4≤t≤t1t−1t−3tctct+1:t=1+t1q−tq−t1+q−tctct+1:2+t1≤t≤qq−t2+q−t1+q−tTable 2
Codes of spokes ofGPq,1 when q is odd.
d.,.c1c3d1c1d1020c2d2111c3d3202ctdt:4≤t≤t1t−1t−3t−1ctdt:t=1+t11+q−tq−t1+q−tctdt:t=2+t11+q−t2+q−t1+q−tctdt:3+t1≤t≤q1+q−t3+q−t1+q−tTable 3
Codes of inner edges ofGPq,1 when q is odd.
d.,.c1c3d1d1d2120d2d3211dtdt+1:3≤t≤t1tt−2t−1dtdt+1:t=1+t11+q−t1+q−tq−tdtdt+1:2+t1≤t≤q1+q−t3+q−tq−tCase 2.
Whenq is even.
We definet2=q/2. The induced subgraph W by W has only one edge, and the codes of all the edges of GPn,1 are given in Tables 4–6.We observe that codes of all the edges in both cases are distinct. So, W is 1-size edge resolving set for GPq,1. Hence, we have β1seGPq,1≤3. Now, we will compute upper bound for the size 2 edge metric dimension of generalized Petersen graphs GPq,1.Table 4
Codes of outer edges ofGPq,1 when q is even.
d.,.c1c3d1c1c2011ctct+1:2≤t≤3t−10ictct+1:4≤t≤t2t−1t−3tctct+1:t=1+t2q−tq−t−11+q−tctct+1:t=2+t2q−t1+q−t1+q−tctct+1:3+t2≤t≤qq−t2+q−t1+q−tTable 5
Codes of spokes ofGPq,1 when q is even.
d.,.c1c3d1c1d1020c2d2111c3d3202ctdt:4≤t≤t2+1t−1t−3t−1ctdt:t=2+t2t2−1t2−1t2−1ctdt:3+t2≤t≤q1+q−t3+q−t1+q−tTable 6
Codes of inner edges ofGPq,1 when q is even.
d.,.c1c3d1d1d2120d2d3211dtdt+1:3≤t≤t2tt−2t−1dtdt+1:t=1+t21+q−tq−tq−tdtdt+1:t=2+t21+q−t2+q−tq−tdtdt+1:3+t2≤t≤qq+1−t3+q−tq−tLemma 7.
For allq≥5,β2seGPq,1≤3.Proof.
LetW=c1,c2,d1 be an edge resolving set for GPq,1. Define k=q/2. The codes of all the edges of GPq,1 with respect to W are given Table 7–9.
Fort=1+k, the code of outer edge ctct+1 will be cWctct+1=q−t,q−t,q−t+1 when q is even and cWctct+1=q−t,q−t+1,q−t+1 when q is odd.
It seems that codes of all the edges are distinct. So,W is size 2 edge resolving set for GPq,1. Hence, we have β2seGPq,1≤3.
In the next lemma, we will give the lower bound for thek-size edge metric dimension of GPq,1 for k=1,2.Table 7
Codes of outer edges ofGPq,1.
d.,.c1c2d1c1c2011ctct+1:2≤t≤kt−1t−2tctct+1:k+2≤t≤qq−t1+q−t1+q−tTable 8
Codes of spokes ofGPq,1.
d.,.c1c2d1c1d1010c2d2:2≤t≤k+1t−1t−2t−1ctdt:k+2≤t≤qq−t+1q−t+2q−t+1Table 9
Codes of inner edges ofGPq,1.
d.,.c1c2d1d1d2110dtdt+1:2≤t≤ktt−1t−1dtdt+1:k+2≤t≤qq−t+1q−t+2q−tLemma 8.
For allq≥7, we have βkseGPq,1≥3.Proof.
First, we will show that there is no edge resolving set ofGPq,1 consisting of two vertices. Contrarily, we suppose that X=A,B be a set having two vertices of VGPq,1. Then, we have the following three possibilities.Case 1.
When both the verticesA and B are from the principal cycle.
Let us fix a vertex, sayA=c1, then B is any other vertex ct2≤t≤q. For 2≤t≤k−1, we have cWcq−1cq=cWd1dq=1,t. For t=k, we have cWc1cq=cWc1d1=0,k−1. For t=k+1, we have cWc2c3=cWcq−1cq=1,k−2 when q is even, while cWc2d2=cWd1dq=1,k−1 when q is odd. For t=k+2, we have cWc2c3=cWd1dq=1,k−1 when q is even, while cWc2c3=cWd1d2=1,q−t+2 when t is odd. For k+3≤t≤q, we have cWc2c3=cWd1d2=1,q−t+2.Case 2.
When bothA and B are the inner vertices.
Let us fix a vertex, sayA=d1, then B is any other vertex vt2≤t≤q.
For2≤t≤k, we have cWd1dq=cWc1d1=0,t−1. For t=1+k, we have cWd3d4=cWdq−2dq−1=2,k−3 when q is even, while cWd3d4=cWcq−1dq−1=2,k−3 when q is odd. For k+2≤t≤q, we have cWc1d1=cWd1d2=0,q−t+1.Case 3.
WhenA is any vertex from the principal cycle and B is any inner vertex.
Let us fix a vertex, sayA=c1; then, B is any inner vertex dt1≤t≤q. For t=1, we have cWc1cq=cWc1c2=0,1. For 2≤t≤k, we have cwc1c2=cWc1d1=0,t−1. For t=1+k, we have cwc1cq=cWc1d1=0,q−t+1 when q is odd; however, cwc1c2=cWc1d1=0,t−1 when q is even. For k+2≤t≤q, we have cWc1cq=cWc1d1=0,q−t+1.
From the above three cases, we conclude that(6)βeGPq,1≥3.
So, there does not exist a size 1 and size 2 edge resolving set of cardinality 2 inGPq,1. Therefore, it yields that βkseGPq,1≥3, for the value of k=1,2.
Lemmas6–8, we conclude the following main result.Theorem 1.
For allq≥7, we have βkseGPq,1=3 when k=1,2.


## 6. Bounds and Some Realizable Results onβkseG

From the earlier discussion, one fundamental question arises. Is thek+1 size edge metric dimension strictly greater than the k-size metric dimension? To answer this question, we consider following two examples.Consider the graphsG1 and G2 which are depicted in Figure 3. It can be observed that the set W=t5,t6,t7 is an edge resolving set for G1 and the cardinality of set W is minimum. Moreover, W1=t1,t5,t6,t7 is a 1ser-set, W2=t5,t2,t6,t7 is a 2ser-set, and W3=t1,t2,t5,t7 is a 3ser-set for G1. Thus, βeG1=3 and β1seG1=β2seG1=β3seG1=4.Figure 3
The graphsG1 and G2.While, for the graphG2≅GP5,1, the set S=p1,p2,q1 is an edge resolving set of the minimum cardinality. Here, the outer vertices are p1,…,p5 and the inner vertices are q1,…,q5. It can be easily seen that the sets S1, S2, and S3 are 1ser-set, 2ser-set, and 3ser-set, respectively, where S1=p1,p2,p4, S2=q1,q2,p1, and S3=p1,p2,p3,p4. Thus, βeGP5,1=β1seGP5,1=β2seGP5,1=3 and β3seGP5,1=4.From these two examples, it can be observed that ifβkseG exists for 1≤k≤t in a nontrivial connected graph G of order n, then(7)βeG≤β1seG≤β2seG≤β3seG………≤βtseG.However, the following example shows that the above inequality is not true, in general.Example 1.
LetG be a graph which is constructed from two graphs C5 and K4. The vertex set of G is l1,l2,l3,l4,l5,p1,p2 and the edge set is l1l2,l2l3,l3l4,l4l5,l5l1,l1p1,p1p2,p2l5,p1l5,p2l1, as shown in Figure 4. The set A=l1,l5,p1,p2 is a 6ser-set of G. However, there is no such set B which resolves all the edges of graph G with cardinality 4 and B has size 5. So, we take the set W=l1,l2,l3,l5,p1 is a 5ser-set of the minimum cardinality for G. Hence, β5seG>β6seG.
Next, we characterize some realizable results for1ser-set and 2ser-set in graphs.Figure 4
The graph in whichβ5seG>β6seG.Theorem 2.
For a nontrivial, simple, and connected graphG of order t, we have β1seG=t if and only if G=P2.Proof.
Lemma1 implies that if G=P2, then β1seG=2=t. Conversely, assume that G be a connected graph of order t≥2 and β1seG=t. Since the induced subgraph W has only one edge, therefore it is obvious that |W|=|VG|=2. Thus, G is a path graph of order 2.
The following result on the complete graphKs was presented in [1].Lemma 9 (see [1]).
For any integers≥2, βeKs=s−1.Theorem 3.
LetGbe a complete graph of orders≥3, thenβ1seGexists iffG=K3. Moreover,β1seG=2.Proof.
One can observe that ifG is a complete graph K3, then the result holds. Conversely, let G be a complete graph of order s≥4. Now, by Lemma 7, the induced subgraph G has more than one edge. Therefore, β1seG does not exist. Hence, the proof is complete.Theorem 4.
LetG be a nontrivial connected graph of order s≥3, then β1seG=s−1 if and only if G=P3 or G=K3≃C3.Proof.
LetG=P3 or G=K3≅C3. From Lemmas 1 and 2, and Theorem 5, we have β1seG=2=s−1. Contrarily, assume that G be a connected graph of order s≥3 and β1seG=s−1. For s=3, it is simple to prove that G=P3 or G=K3≃C3. Now, we will prove the result for s≥4. For this, we suppose W be a 1ser-set of G of cardinality s−1. It is easy to see that VG−W=1; therefore, the induced subgraph has surely more than one edge in it (G is a connected graph). It yields that W=VG=3. Thus, G=P3 or G=K3≃C3.Remark 2.
The two size edge resolving sets exist in complete bipartite graphG=Ks,t if and only if G∈K2,1,K2,2,K2,3,K3,1. Moreover, β2seK2,1=3=s+t, β2seK3,1=β2seK2,2=3=s+t−1, andβ2seK2,3=3=s+t−2.Theorem 5.
For a simple and connected graphGof orders≥3,β2seG=sif and only ifG=P3≅K2,1.Proof.
By Lemma1 and Remark 2, if G=P3≅K2,1, then β2seG=3=s. Conversely, suppose that β2seG=s for a connected graph G of order s≥3. Let W be a 2ser-set of order s. Since G is a connected graph and induced subgraph has two edges which implies that W=VG=3, thus G=P3≃K2,1.
Now, we study a sufficient condition for a pairq,p of positive integers to be realizable as the order and the k-size edge metric dimension of a connected graph, respectively.Theorem 6.
For a pairq,p of positive integers with q≥p≥2, there exists a connected graph G of order q and βkseG=p, where k+1≤p≤q.Proof.
Fork≥1, we consider the following two cases according to the choice of p.(i)
Forp=k+1, let G=Pq be a path graph of order q≥2. Lemma 1 implies that βksePq=p, where 1≤k≤q−1. For p=q, let G=Cq be a cycle of order q≥3, then it seems that βkseCq=p, where k=p.(ii)
Fork+2≤p≤q−1, let H be a connected graph of order q≥6 obtained from paths Pq−p:e1,e2,e3…,eq−pq−p≥2, where p≥4, Pk+1:f1,f2,f3,…,fk+11≤k≤p−3 and p−k−1 vertices g1,g2,…,gp−k−1 with gi∼e1 and cj∼e1 for 1≤i≤k+1 and 1≤j≤p−k−1, as shown in Figure 5. Firstly, we prove that βkseH≤p. For this, let S=f1,f2,…,fk+1,g1,g2…,gp−k−1⊆VH. Since, the subgraph induced by S has k edges and cSeiei+1=i,i,i,…,i,i for each 1≤i≤q−p−1, it implies that S is a kser-set for G, and hence βkseH≤l. Now, to prove βkseG≥l, we suppose contrarily that βkseG≤l−1 and W be a kser-set for H with |W|≤p−1. If W contains at least p−k−2 vertices from C, say gi1≤i≤p−k−2 and the size of W is equal to k, it follows that W=VPk+1∪g1,g2,…,gp−k−2. However, still we have cWe1e2=cWe1gp−k−1=1,1,1,…,1,1. Thus, βkseH≥p which yields that βksH=p.Figure 5
The graphH.


## 7. Conclusions

In this work, we have introduced a new variant, namely, thek-size edge metric dimension of graphs, and initiated its study by finding the k-size edge metric dimension of several well-known classes of graphs. We have characterized the graphs having k-size edge metric dimension n and n−1. Moreover, we have computed the k-size edge metric dimension of the Cartesian product graphs Pm□Pn and Pm□Cn for the values of k=1,2,3. In addition, we have proved that the k-size edge metric dimension of generalized Petersen graphs GPq,1 is 3 for the values of k=1,2. Some realizable results on k-size edge resolvability are also presented in this paper [8–12].



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*Source: 1023175-2020-12-31.xml*