( k, l ) -Anonymity in Wheel-Related Social Graphs Measured on the Base of k -Metric Antidimension

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Introduction
Since 2016, a novel privacy measure, "the (k, l) anonymity," is defined and used, for the sake of a social graph confrontation from various active attacks, in connection with the concept of k-metric antidimension. Trujillo-Rasua and Yero defined, studied in detail, and promoted the idea of k-metric antidimension, which provides a basis for the privacy measure (k, l)-anonymity [7]. ey defined this privacy measure as follows: " e (k, l)-anonymity for a social graph G will be preserved according to active attacks if the k-metric antidimension of G is bounded above by l for the least positive integer k, where l is an upper bound on the expected number of attacker nodes." Accordingly, it can be seen that having a k-antimetric generator (a set defining the k-metric antidimension) as the set of attacker nodes, the probability of unique identification of other nodes by an adversary in a social graph is less than or equal to (1/k).
Besides, providing many significant theoretical properties of the k-metric antidimension of graphs, Trujillo-Rasua and Yero also supplied the k-metric antidimension of complete graphs, paths, cycles, complete bipartite graphs, and trees [7]. is significant work of Trujillo-Rasua and Yero attracted many researchers to work on this idea, and therefore, the literature has been updated with the following remarkable contributions up till now: (i) Trujillo-Rasua and Yero further contributed by characterizing 1-metric antidimensional trees and unicyclic graphs [8] (ii) Mauw et al. contributed by providing a privacypreserving graph transformation, which improves privacy in social network graphs by contracting active attacks [6] (iii)Čangalović et al. contributed by considering wheels and grid graphs in the context of the k− metric antidimension [1] (iv) DasGupta et al. contributed by analyzing and evaluating privacy-violation properties of eight social network graphs [4] (v) Kratica et al. contributed by investigating the k− metric antidimension of two families of generalized Petersen graphs GP(n, 1) (also called prism graphs) and GP(n, 2) [5] (vi) Zhang and Gao and, later on, Chatterjee et al. contributed by proving that the problem of finding the k-metric antidimension of a graph is, generally, an NP-complete problem [3,9] Inspired by all these contributions and, particularly, motivated by the work done byČangalović et al. on wheel graphs, we place our contribution by extending the study of (k, l)-anonymity privacy measure based on k-metric antidimension towards four families of wheel-related social graphs.

Basic Works
Let G � (V(G), E(G)) be a simple and connected graph. We denote two adjacent vertices x and y by x ∼ y and nonadjacent by x ≁ y in G. Two vertices of G are said to be neighbors of each other if there is an edge between them. e (open) neighborhood of a vertex x in G is N(x) � y ∈ V(G) : y ∼ x ∈ E(G)}. e neighborhood N(x) is closed if it includes x and is denoted by N [x]. e number of vertices adjacent with a vertex x is called its degree and is denoted by d(x). e maximum degree in G is Δ � max x∈V(G) d(x). e metric on G is a mapping d : V(G) × V(G) ⟶ Z + ∪ 0 { } defined by d(x, y) � l, where l is the length of the number of edges in the shortest path between vertices x and y in G. A vertex u of G identifies a pair (x, y) of vertices in G if d(x, u) ≠ d(y, u). e sum G + H of two graphs G and H is obtained by joining each vertex of G with every vertex of H. We refer the book in [2] for nonmentioned graphical notations and terminologies used in this paper.
Let S � s 1 , s 2 , . . . , s t ⊆ V(G) be an ordered set. en, the metric code, or simply code, of a vertex u ∈ V(G) with respect to S is the t-vector c S (u) � (d(u, s 1 ), d(u, s 2 ), . . . , d (u, s t )). A chosen set S of vertices of G unique identifies each pair (x, y) of vertices in G if c S (x) ≠ c S (y). e following concepts are defined by Trujillo-Rasua and Yero in [7]: (i) A set S of vertices of G is called a k− antimetric generator (k-antiresolving set) for G if k is the largest positive integer such that k vertices of G, other than the vertices in S, are not uniquely identified by S; i.e., for every vertex w ∈ V(G) − S, there exist at least k − 1 different vertices u 1 , ii) e cardinality of the smallest k-antimetric generator for G is called the k-metric antidimension of G, denoted by adim k (G), and such a smallest generator is known as k-antimetric basis of G (iii) If k is the largest positive integer such that G has a k-antimetric generator, then G is said to be k-metric antidimensional graph If S is a set of vertices of a graph G, then it has been defined as a relation on V(G) − S according to the vertices having equal metric codes with respect to S as follows. [5,7]. Let S ⊂ V(G) be a set of vertices of a connected graph G and let ρ S be a relation on

Equivalence Relation and Classes
is relation is an equivalence relation and partitioned V(G) − S into classes, say S 1 , . . . , S m , called the equivalence classes corresponds to the relation ρ S . Accordingly, we get the following useful property from [5].
Remark 1 (see [5]). For a fixed integer k ≥ 1, a set S is a k-antimetric generator for G if and only if min m i�1 |S i | � k, where each S i , 1 ≤ i ≤ m, is an equivalence class defined by the relation ρ S .

Wheel-Related Social Graphs
In this section, we consider five wheel-related social graphs. e (k, l)-anonymity of one of them, called a wheel graph, has been measured previously in [1], by investigating its k-metric antidimension. Here, we focus to investigate the k-metric antidimension of other four graphs. For n ≥ 3, a wheel graph is W 1,n � K 1 + C n , where K 1 is the trivial graph having only one vertex ], and C n is a cycle graph with where the indices greater than n or less than 1 will be taken modulo n. Each edge ] ∼ v i is called a spoke in a wheel graph. One such graph is depicted in Figure 1.
In 2018,Čangalović et al. supplied the following investigations.
Observation 1 (see [1]) Theorem 1 (see [1]). For all n ≥ 6, e rest of the section is aimed to investigate the k-metric antidimensions of Jahangir graphs, helm graphs, flower graphs, and sunflower graphs.

Jahangir Graphs.
For n ≥ 2, a Jahangir (Gear) graph, J 2n , is obtained from a wheel graph W 1,2n � K 1 + C 2n by deleting alternating spokes from the wheel.
en, the vertex set of a Jahangir graph is where the indices greater than n or less than 1 will be taken modulo n. Figure 2 depicts graphical view of one Jahangir graph.
e following observation is easy to verify for n � 2, 3, and 4.

Observation 2
and adim k ( For all values of n ≥ 5, the following result provides the k-metric antidimension of Jahangir graphs.

Theorem 2.
For n ≥ 5, let J 2n be a Jahangir graph. en, Proof. First of all, it is worthy to note that N(]) � V(C 2n ) and , for any u i ∈ U. Now, we need to discuss the following seven claims.
{ } is an n-antimetric generator for J 2n . Note that c S (x) � (1), for all x ∈ V, and c S (y) � (2), for all y ∈ U. According to the relation ρ S , there are only two equivalence classes each has cardinality n. So, the result followed by Remark 1. Claim 2: every singleton subset of V is a 3-antimetric generator for J 2n .
Hence, the relation ρ S supplies three equivalence classes Figure 2: One Jahangir graph J 8 .
Journal of Mathematics min 3 i�1 |S i | � 3, and hence, S is 3-antimetric generator, by Remark 1. Claim 3: every singleton subset of U is a 2-antimetric generator for J 2n . Let S � u i ⊂ U; then, metric codes of the vertices are Clearly, we receive four equivalence classes according to the relation ρ S , Hence, min 4 i�1 |S i | � 2, and S is a 2− antimetric generator, by Remark 1. Claim 4: every 2 element subset of V(J 2n ) is either 1-antimetric generator or 2-antimetric generator for J 2n . Let S be a 2-element subset of V(J 2n ). en, we have the following two cases to discuss. Case 1 (S contains ]): here, we have two subcases.
So, the equivalence classes corresponds to the re- Here, min 3 i�1 |S i | � 2, which implies that S is a 2-antimetric generator, by Remark 1.
Due to the above metric coding, it is clear that we find four equivalence classes according to the relation ρ S , which are us, min 4 i�1 |S i | � 2, which implies that S is a 2− antimetric generator, by Remark 1. Case 2 (S does not contain ]): again, we have three subcases to discuss.
y ∈ U, such that d(y, x) � 2, has the unique metric code from the set (1, 3), (3, 1) { } with respect to S. If no vertex from U is a common neighbor of v and v ′ , then the vertex ] has the unique metric code (1, 1) with respect to S. Subcase 2.2: let S ⊂ U and S � u, u ′ . en, either d(u, u ′ ) � 2 or d(u, u ′ ) � 4. In the former case, a vertex v ∈ V, such that v ∈ N(u) ∩ N(u ′ ), has the unique metric code (1, 1). In the later case, we have two possibilities. If there is a vertex x ∈ U such that d(x, u) � 2 � d(x, u ′′ ), then a vertex y ∈ U, with d(y, u) � 2 and d(y, u ′ ) � 4, has the unique metric code (2, 4) with respect to S. If there is no such x in U, then the vertex ] has the unique metric code (2, 2) with respect to S.
Here, we have two possibilities. If one of the neighbors x and y of v, say x, has the property that d(x, u) � 2, then the neighbor y of v has the unique metric code (4, 1) with respect to S. If d(x, u) � 4 � d(y, u), then the vertex ] has the unique metric code (2, 1) with respect to S. In the former case, v is a one neighbor of u from V, and the other neighbor of u from V has the unique metric code (1, 2) with respect to S.
In each possibility of these subcases, the relation ρ S proposes at least one singleton equivalence class, which follows that min i |S i | � 1. Hence, S is an 1− antimetric generator, by Remark 1.
, the following two cases are need to be discussed for x.
(16) So, the equivalence classes corresponds to the relation ρ A′ are Note that min 5 i�1 |S i | � 2, so Remark 1 yields the required result. 4 Journal of Mathematics Hence, the equivalence classes corresponds to re- , then again we have two cases.
In both the cases, we get at least one singleton equivalence class t { } with respect to the relation ρ A′ , which implies that min i |S i | � 1. Hence, A ′ is a 1− antimetric generator, by Remark 1.
Hence, we have the equivalence classes In each possibility, the given metric code is unique, which provides a singleton equivalence class according to the relation ρ B . Hence, min i |S i | � 1, and B is a 1− antimetric generator, by Remark 1. Claim 7: every set S ⊆ V(J 2n ) of cardinality t ≥ 3 is a 1− antimetric generator for J 2n , except the sets A ′ and B discussed in Claims 5 and 6, respectively.
If S contains the vertex ], then there exists a vertex g ∈ U (or g ∈ V) such that g is a neighbor of some element in S, and c S (g) ≠ c S (g ′ ), for any g ′ ∈ V(J 2n ) − g . If S does not contain the vertex ], then ] has the unique metric code with respect to S. In both the cases, we get a singleton equivalence class according to the relation ρ S . Hence, min i |S i | � 1, and Remark 1 implies that S is a 1-antimetric generator.
All these claims conclude the proof with the following points: there does not exist a k-antimetric generator for J 2n . (ii) Claims 1, 2, and 3 provide that adim 1 (J 2n ) ≥ 2.

Helm
Graphs. For n ≥ 3, a helm graph, H n , is obtained from a wheel graph W 1,n � K 1 + C n by attaching one leaf (a vertex of degree one) with each vertex of the cycle C n . Let U � u 1 , u 2 , . . . , u n be the set of leaves; then, the vertex set of a helm graph is V(H n ) � V(W 1,n ) ∪ U, and its edge set is where the indices greater than n or less than 1 will be taken modulo n. One helm graph is shown in Figure 3. It is an easy task to verify the following observation.
Theorem 3. For n ≥ 7, let H n be a helm graph. en, Proof. It is significant to keep in hand the neighborhoods Now, we have to discuss the following eight claims. (2), for all y ∈ U. ere are only two equivalence classes S 1 � V and S 2 � U, both of cardinality n, according to the relation ρ S . Hence, min 2 i�1 |S i | � n, which implies that S is an n-antimetric generator, by Remark 1. Claim 2: every single leaf form a 1-antimetric generator for H n . Let S � u i be a set of one leaf u i ∈ U. en, , and no other vertex of H n has the code similar to v i . It follows that there exists a singleton equivalence class due to the relation ρ S . Accordingly, Remark 1 refers that S is an 1− antimetric generator.
Claim 3: every singleton subset of V is a 4-antimetric generator for H n .
us, the relation ρ S produces three equivalence classes , and S is a 4antimetric generator for H n , by Remark 1. Claim 4: the set S ′ � u i , w is a 3-antimetric generator for H n whenever w ∈ N(u i ). Otherwise, S ′ is a 1-antimetric generator. Whenever w ∈ N(u i ), we have w � v i , and the metric codes with respect to S ′ are e equivalence classes corresponds to the relation ρ S′ are has the metric code which is not similar to the metric code of any other vertex of H n . is creates at least one singleton equivalence class v i in accordance with the relation ρ S′ , which implies that S ′ is a 1-antimetric generator for H n , by Remark 1. Claim 5: every 2− element subset of V(H n ), except the set S ′ of Claim 4, is a 1-antimetric generator for H n . Let S be a 2− element subset of V(H n ) and S ≠ S ′ . en, we discuss the following two cases: Case 1 (S does not contain ]): let S ⊂ U. When S � u i , u i+2 , the vertex u i+1 has the unique metric code (3, 3) with respect to S. When S � u i , u i− 2 , the vertex u i− 1 has the unique metric code (3, 3) with respect to S. Otherwise, the vertex ] has the unique metric code (2, 2) with respect to S. Next, we let S ⊂ V. When S � v i , v i+2 , the vertex v i− 1 has the unique metric code (1, 2) with respect to S. Otherwise, the vertex ] has the unique metric code (1, 1) with respect to S. Case 2 (S contains ]): let S � v i , ] ; then, the leaf u i ∈ N(v i ) has the unique metric code (1, 2) with respect to S.
Each possibility in both the cases yields at least one singleton equivalence class according to the relation ρ S , which implies that S is a 1-antimetric generator, by Remark 1.  Journal of Mathematics So, there are four equivalence classes with respect to the relation ρ E . at is, min 4 i�1 |S i | � 2, and Remark 1 assists that E is a 2antimetric generator. Claim 7: for even values of n ≥ 8, the set S � ], v i , u i , v i+(n/2) , u i+(n/2) ⊂ V(H n ) is a 2-antimetric generator for H n . Note the metric coding of the vertices with respect to S is as follows: Hence, the classes according to the relation ρ S are Here, min 6 i�1 |S i | � 2, which implies that S is a 2-antimetric generator, by Remark 1. Claim 8: any subset of V(H n ) of cardinality t ≥ 3 is a 1antimetric generator for H n , except the sets E and S considered in Claims 6 and 7, respectively.
Let W be a subset of V(H n ) with |W| � t ≥ 3 and W ≠ E, S. en, note the following two possibilities: In both the possibilities, we get at least one singleton equivalence class according to the relation ρ W . us, min i |S i | � 1, and Remark 1 provides that S is a 1-antimetric generator. e proof will reach to its end by discussing the following points on the base of formerly discussed claims: (i) For k ∈ 5, 6, . . . , n − 1 { }, there does not exist a k-antimetric generator for H n . (ii) We find a 1-antimetric generator for H n of (1) cardinality 1 due to Claim 2, (2) cardinality 2 due to Claims 4 and 5, and (3) cardinality t ≥ 3 due to Claim 8. Since a 1-antimetric generator for H n of cardinality 1 is the smallest one, so adim 1 (H n ) � 1. (iii) Claim 6 assures the existence of a 2-antimetric generator for H n of cardinality 3 for all values of n, while Claim 7 assures the existence of a 2-antimetric generator for H n of cardinality 5 just for even values of n. Moreover, no singleton set or 2-element set of vertices in H n is a 2-antimetric generator for H n due to Claims 1 to 5. It follows that adim 2 (H n ) � 3. (iv) We receive a 3− antimetric generator of cardinality 2 from Claim 4, and no singleton set is a 3-antimetric generator for H n due to Claims 1 to 3, which implies that adim 3 (H n ) � 2.

Flower Graphs.
For n ≥ 3, a flower graph, F n , is obtained from a helm graph H n by joining its each leaf u i to the vertex ] of K 1 . Accordingly, the vertex set of a flower graph is V(F n ) � V(H n ), and its edge set is E(F n ) � E(H n ) ∪ ] ∼ u i ; 1 ≤ i ≤ n , where the indices greater than n or less than 1 will be taken modulo n. Figure 4 provides graphical appearance of one flower graph. e following observation is easy to understand for the flower graph F 3 .

Observation 4
Proof. e following listed neighborhoods of the vertices of F n will be used in the proof: N(]) � V ∪ U and N(v i ) � ], v i+1 , v i− 1 , u i , for any v i ∈ V, and N(u i ) � ], v i , for any u i ∈ U. We have to discuss the following nine claims to prove the required result.

Claim 1: the set S � ]
{ } is a 2n-antimetric generator for F n .

Journal of Mathematics
Note that c S (x) � (1), for all x ∈ V(F n ). So, the only one equivalence class of cardinality 2n is produced by the relation ρ S . Hence, S is a 2n-antimetric generator. Claim 2: every singleton subset of U is a 2− antimetric generator for F n . Let S � u i ⊂ U for any fixed 1 ≤ i ≤ n; then, c S (x) � (1), for all x ∈ N(u i ), and c S (y) � (2), for all e relation ρ S creates two equivalence classes It follows that min 2 i�1 |S i | � 2, and Remark 1 proposes that S is a 2-antimetric generator. Claim 3: every singleton subset of V is a 4-antimetric generator for F n . Let S � v i ⊂ V for any fixed 1 ≤ i ≤ n; then, c S (x) � (1), for all x ∈ N(v i ), and c S (y) � (2), for all us, we get two equivalence classes S 1 � N(v i ) and S 2 � V(F n ) − N[v i ] by the relation ρ S with min 2 i�1 |S i | � 4. Hence, S is a 4-antimetric generator, by Remark 1. Claim 4: the set W � v i , ] ⊂ V(F n ) is a 3-antimetric generator for F n . Note the metric codes with respect to W is as follows: Here, the equivalence classes obtained through the relation ρ W are Hence, min 2 i�1 |S i | � 3, and Remark 1 implies that W is a 3-antimetric generator.
e metric codes with respect W ′ are c W′ (x) � (1, 1), for all x ∈ N(u), c W′ (y) � (1, 2), for all y ∈ N(v i ) − N(u), and c W′ (z) � (2, 2), for all Accordingly, three equivalence classes S 1 � N(u), S 2 � N(v i ) − N(u), and [u]) are generated by the relation ρ W′ with min 3 i�1 |S i | � 2. erefore, Remark 1 provides that W ′ is a 2− antimetric generator. Claim 6: any 2-element set, S ⊆ V(F n ), is a 1-antimetric generator for F n , except the sets W and W ′ discussed in Claims 4 and 5, respectively. We discuss the following two possibilities: In both the possibilities, we receive at least one singleton equivalence class, in accordance with the relation ρ S , which implies that min i |S i | � 1. Hence, Remark 1 yields that S is a 1-antimetric generator.
e metric coding with respect to M is are the equivalence classes produced by the relation ρ M . Here, min 2 i�1 |S i | � 2, which implies that M is a 2-antimetric generator, by Remark 1.
, v i+2 , and we have the metric codes of the vertices with respect to M ′ as follows: then a neighbor of f lying in U has the unique metric code (2, 1, 1) with respect to M ′ . If f � v i+2 or v i− 2 , then v i+1 or v i− 1 , respectively, has the unique metric code (1, 1, 1) with respect to M ′ . In either cases, we obtain at least one singleton equivalence class according to the relation ρ M′ , which implies that M ′ is a 1-antimetric generator, by Remark 1. Claim 9: any set S ⊆ V(F n ) of cardinality t ≥ 3 is a 1− antimetric generator for F n , except the sets M and M ′ discussed in Claims 7 and 8, respectively. We discuss the following two cases: Case 1 (S does not contain ]): in this case, the vertex ] has the unique metric code with respect to S In both the cases, we get at least one singleton equivalence class according to the relation ρ S , which implies that S is a 1− antimetric generator, by Remark 1.
We conclude the proof by discussing the following points using preceding claims: (i) For k ∈ 5, 6, . . . , 2n − 1 { }, there does not exist a k-antimetric generator for F n . (ii) We get an 1-antimetric generator for F n of (1) cardinality 2 by Claim 6 and (2) cardinality t ≥ 3 by Claims 8 and 9. Furthermore, no singleton set possesses the property of 1− antimetric generator in F n , by Claims 1 to 3. It follows that adim 1 (F n ) � 2. (iii) For F n , Claim 2 promises the existence of a 2-antimetric generator of cardinality 1, Claim 5 promises the existence of a 2-antimetric generator of cardinality 2, and Claim 7 promises the existence of a 2-antimetric generator of cardinality 3. All these promises conclude that adim 2 (F n ) � 1. (iv) ere exists a 3-antimetric generator for F n of cardinality 2 due to Claim 4, and a 3-antimetric generator of cardinality 3 due to Claim 8. us, Claims 1 to 3 conclude that adim 3 (F n ) � 2.

Sunflower Graphs.
For n ≥ 3, a sunflower graph, SF n , is obtained from a wheel graph W 1,n � K 1 + C n by attaching one vertex u i to every two consecutive vertices of the cycle C n . Let U � u 1 , u 2 , . . . , u n ; then, the vertex set of a sun flower graph is V(SF n ) � V(W 1,n ) ∪ U and its edge set is where the indices greater than n or less than 1 will be taken modulo n.
A graphical preview of this graph is displayed in Figure 5. e following observation is an easy exercise to understand.
, for any u i ∈ U, of the vertices in SF n are useful to discuss the following nine claims.

Claim 1: the set S � ]
{ } is an n-antimetric generator for SF n . Note that c S (x) � (1), for all x ∈ V, and c S (y) � (2), for all y ∈ U. us, there are two equivalence classes V and U according to the relation ρ S , and each class has n elements. Hence, Remark 1 yields that S is an n-antimetric generator. Claim 2: every singleton subset S of V is a 5− antimetric generator for SF n .
(43) erefore, the equivalence classes, corresponding to the relation ρ S , are i+1 . It follows that min 3 i�1 |S i | � 5, and S is a 5-antimetric generator, by Remark 1. Claim 3: every singleton subset S of U is a 2− antimetric generator for SF n . Let S � u i ⊂ U, for any fixed 1 ≤ i ≤ n. en, Figure 5: One sunflower graph SF 8 .

Journal of Mathematics
We have four equivalence classes S 1 � N(u i ), in accordance with the relation ρ S . us, min 4 i�1 |S i | � 2, which implies that S is a 2-antimetric generator, by Remark 1. Claim 4: the set S � v i , ] ⊂ V(SF n ) is a 2-antimetric generator for SF n . e metric coding with respect to S is listed as follows: So, the relation ρ S supplies five equivalence classes i�1 |S i | � 2, and S is a 2-antimetric generator, by Remark 1. Claim 5: the set S � u i , ] ⊂ V(SF n ) is a 2-antimetric generator for SF n . We have the following metric coding with respect to S: c S (t) � (1, 1), for all t ∈ N(u i ): So, the equivalence classes, in accordance with the relation ρ S , are Here, min 6 i�1 |S i | � 2, which implies that S is a 2-antimetric generator, by Remark 1.
{ } is a 1− antimetric generator for SF n . We discuss the following three possibilities: N(v), and c S (p) � (1, 1) ≠ c S (p ′ ), for any p ′ ∈ V(SF n ) − p . If d(u, v) � 3, then there is a neighbor q of v from U such that c S (q) � (4, 1) ≠ c S (q ′ ), for any q ′ ∈ V(SF n ) − q . If d(u, v) ≥ 4, then the vertex ] has the unique metric code (2, 1) with respect to S.
In the former case, a vertex u ∈ U, for which d(u, v) � 2 and d(u, v ′ ) � 3, has the unique metric code (2, 3) with respect to S. In the later case, we have two discussions: If such that d(u, v ′′ ) � 2, has the unique metric code from the set (1, 3), (3, 1) { } with respect to S. If no vertex in V is a common neighbor of v and v ′ , then the vertex ] has the unique metric code (1, 1) with respect to S.
(3) Let S ⊂ U; then, c S (]) � (2, 2) is the unique metric code in SF n . In all these possibilities, we get at least one singleton equivalence class according to the relation ρ S , which implies that min i |S i | � 1. Hence, S is a 1-antimetric generator, by Remark 1.
Hence, we have eight equivalence classes S 1 � u i− 1 , u i , in accordance with the relation ρ E . It can be seen that min 8 i�1 |S i | � 2, which yields that E is a 2-antimetric generator, by Remark 1.
Hence, we receive at least one singleton equivalence class due to the relation ρ E , which implies that E is a 1-antimetric generator, by Remark 1.
we have the metric codes with respect to E ′ as follows: erefore, we get 10 equivalence classes S 1 � N(u i ), , and S 10 � U − (S 2 ∪ S 4 ∪ S 6 ∪ S 8 ) in accordance with the relation ρ E′ . It has been observed that min 10 i�1 |S i | � 2, so E ′ is a 2− antimetric generator, by Remark 1. Whenever a ∈ u i− 4 , u i− 3 , u i− 2 , u i− 1 , u i+1 , u i+2 , u i+3 , u i+4 , we have a vertex u ∈ U such that c E′ (u) ≠ c E′ (u ′ ), for any u ′ ∈ V(SF n ) − u { }. Hence, we receive at least one singleton equivalence class by the relation ρ E′ , which implies that min i |S i | � 1. Hence, E ′ is a 1− antimetric generator, by Remark 1. Claim 9: except the sets E, E ′ ⊂ V(SF n ) discussed in Claims 7 and 8, respectively, each set S ⊆ V(SF n ) of cardinality k ≥ 3 is a 1− antimetric generator for SF n .
We have to discuss the following two cases: Case 1 (S contains ]): let |S| ≥ 4 (because the case, when |S| � 3, has been discussed in Claims 7 and 8).
en, there a vertex x ∈ V such that x is a neighbor of some s ∈ S whenever S ∩ V � ∅, or there a vertex x ∈ U such that x is a neighbor of some s ∈ S whenever either S ∩ U � ∅ or S ∩ V ≠ ∅ ≠ S ∩ U, and we get the unique metric code of x with respect to S. Case 2 (S does not contain ]): whenever S ⊆ V or S ⊆ U, the vertex ] has the unique metric code of x with respect to S.
In both the cases, the relation ρ S supplies at least one singleton equivalence class, which yields that min i |S i | � 1. Hence, S is a 1-antimetric generator, by Remark 1.
ese claims complete the proof with the following deductions: (i) ere does not exist a k− antimetric generator for SF n when k ∈ 3, 4, 6, 7, . . . , n − 1 { }. (ii) Claim 6 supplies a 1− antimetric generator for SF n of cardinality 2, and Claims 7 to 9 supply a 1− antimetric generator for SF n of cardinality t ≥ 3. Claims 1 to 3 provide the guaranty of nonexistence of singleton 1− antimetric generator for SF n . It follows that adim 1 (SF n ) � 2. (iii) e existence of a 2− antimetric generator for SF n of cardinalities 1, 2, and 3 is assured by Claim 3, by Claims 4 and 5, and by Claims 7 and 8, respectively. Accordingly, adim 2 (SF n ) � 1. (iv) adim 5 (SF n ) � 1 because of the existence of a 5− antimetric generator for SF n of cardinality 1 in Claim 2. (v) Claim 1 provides an n− antimetric generator for SF n of cardinality 1, which yields that adim n (SF n ) � 1.

Concluding Remarks
is called the radius of G, where ecc(x) is the eccentricity of x. e center of G is a subgraph G[X] induced by the set It has been observed the following useful properties about a k− antimetric dimensional graph in [7].
Remarks 2 (see [7]) (a) If a connected graph is k− metric antidimensional, then 1 ≤ k ≤ Δ (b) If the center of a connected graph is trivial, then it is k− metric antidimensional for some k ≥ 2 It can be easily seen that each wheel-related social graph, considered in this paper, has the trivial center. Remark 2 (a) insures that each of these graphs has k− metric antidimension, for some k ≥ 1, and it must be k− metric antidimensional for some k ≥ 2. So, naturally, it raises the following two questions: Q1. For how many and for which values of k ≥ 1 a wheel-related social graph admits k− metric antidimension? Q2. For which maximum value of k, 2 ≤ k ≤ Δ, a wheelrelated social graph is k− metric antidimensional? e results ofČangalović et al., proved in [1] and listed in Observation 1 and eorem 1, were the pioneers to address the answers of questions Q1 and Q2. ese results revealed that (1) when n � 3, 5, a wheel graph W 1,n admits k− metric antidimension for three values of k ∈ 1, 2, n { } and (2) when n � 4 and for all n ≥ 6, W 1,n admits k− metric antidimension for four values of k ∈ 1, 2, 3, n { }. It follows that W 1,n is Δ− metric antidimensional.
To extend the study of (k, l)− anonymity based on the k− metric antidimension, we considered four graphs related to wheel graphs in this article. By investigating their k− metric antidimension, we addressed the answers of questions Q1 and Q2 as follows: From all these results, it can be concluded that each considered wheel-related social graph is Δ− metric antidimensional.
Furthermore, according to the computed k− metric antidimension of wheel-related social graphs, we investigated that each of them meets the (k, l)− anonymity in the following ways (skipping particular cases which can be observed straightforwardly).
Wheel graph: for n ≥ 6 and Δ � n and by eorem 1, we have e (k, l)− anonymity, measured on the base of k− metric antidimension for the maximum value of k � Δ, assures that a user can be reidentified with the probability less than or equal (1/Δ) by a rival controlling only single attacker node ] in every considered wheel-related social graph. It is remarkably interesting to leave the following conjecture for the readers. Data Availability e figures, tables, and other data used to support this study are included within the article.

Conflicts of Interest
e authors declare that there are no conflicts of interests regarding the publication of this paper.