Abstract
For the study and valuation of social graphs, which affect an extensive range of applications such as community decision-making support and recommender systems, it is highly recommended to sustain the resistance of a social graph to active attacks. In this regard, a novel privacy measure, called the -anonymity, is used since the last few years on the base of -metric antidimension of in which is the maximum number of attacker nodes defining the -metric antidimension of for the smallest positive integer . The -metric antidimension of is the smallest number of attacker nodes less than or equal to such that other nodes in cannot be uniquely identified by the attacker nodes. In this paper, we consider four families of wheel-related social graphs, namely, Jahangir graphs, helm graphs, flower graphs, and sunflower graphs. By determining their -metric antidimension, we prove that each social graph of these families is the maximum degree metric antidimensional, where the degree of a vertex is the number of vertices linked with that vertex.
1. Introduction
Since 2016, a novel privacy measure, “the anonymity,” is defined and used, for the sake of a social graph confrontation from various active attacks, in connection with the concept of -metric antidimension. Trujillo-Rasua and Yero defined, studied in detail, and promoted the idea of -metric antidimension, which provides a basis for the privacy measure –anonymity [7]. They defined this privacy measure as follows:
“The-anonymity for a social graphwill be preserved according to active attacks if the-metric antidimension of is bounded above byfor the least positive integer, whereis an upper bound on the expected number of attacker nodes.”
Accordingly, it can be seen that having a -antimetric generator (a set defining the -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 .
Besides, providing many significant theoretical properties of the -metric antidimension of graphs, Trujillo-Rasua and Yero also supplied the -metric antidimension of complete graphs, paths, cycles, complete bipartite graphs, and trees [7]. This 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 -metric antidimensional trees and unicyclic graphs [8](ii)Mauw et al. contributed by providing a privacy-preserving 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 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 metric antidimension of two families of generalized Petersen graphs (also called prism graphs) and [5](vi)Zhang and Gao and, later on, Chatterjee et al. contributed by proving that the problem of finding the -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 by Čangalović et al. on wheel graphs, we place our contribution by extending the study of -anonymity privacy measure based on -metric antidimension towards four families of wheel-related social graphs.
2. Basic Works
Let be a simple and connected graph. We denote two adjacent vertices and by and nonadjacent by in . Two vertices of are said to be neighbors of each other if there is an edge between them. The (open) neighborhood of a vertex in is . The neighborhood is closed if it includes and is denoted by . The number of vertices adjacent with a vertex is called its degree and is denoted by . The maximum degree in is . The metric on is a mapping defined by , where is the length of the number of edges in the shortest path between vertices and in . A vertex of identifies a pair of vertices in if . The sum of two graphs and is obtained by joining each vertex of with every vertex of . We refer the book in [2] for nonmentioned graphical notations and terminologies used in this paper.
Let be an ordered set. Then, the metric code, or simply code, of a vertex with respect to is the -vector . A chosen set of vertices of unique identifies each pair of vertices in if . The following concepts are defined by Trujillo-Rasua and Yero in [7]:(i)A set of vertices of is called a antimetric generator (-antiresolving set) for if is the largest positive integer such that vertices of , other than the vertices in , are not uniquely identified by ; ., for every vertex , there exist at least different vertices such that (ii)The cardinality of the smallest -antimetric generator for is called the -metric antidimension of , denoted by , and such a smallest generator is known as -antimetric basis of (iii)If is the largest positive integer such that has a -antimetric generator, then is said to be -metric antidimensional graph
If is a set of vertices of a graph , then it has been defined as a relation on according to the vertices having equal metric codes with respect to as follows.
2.1. Equivalence Relation and Classes [5, 7]
Let be a set of vertices of a connected graph and let be a relation on defined by
This relation is an equivalence relation and partitioned into classes, say , called the equivalence classes corresponds to the relation .
Accordingly, we get the following useful property from [5].
Remark 1. (see [5]). For a fixed integer , a set is a -antimetric generator for if and only if , where each , , is an equivalence class defined by the relation .
3. Wheel-Related Social Graphs
In this section, we consider five wheel-related social graphs. The -anonymity of one of them, called a wheel graph, has been measured previously in [1], by investigating its -metric antidimension. Here, we focus to investigate the -metric antidimension of other four graphs. For , a wheel graph is , where is the trivial graph having only one vertex , and is a cycle graph with vertices in . Accordingly, the vertex set of this graph is and edge set is , where the indices greater than or less than 1 will be taken modulo . Each edge 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 ,
The rest of the section is aimed to investigate the -metric antidimensions of Jahangir graphs, helm graphs, flower graphs, and sunflower graphs.
3.1. Jahangir Graphs
For , a Jahangir (Gear) graph, , is obtained from a wheel graph by deleting alternating spokes from the wheel. Let , where and . Then, the vertex set of a Jahangir graph is and its edge set is , where the indices greater than or less than 1 will be taken modulo . Figure 2 depicts graphical view of one Jahangir graph.
The following observation is easy to verify for , and 4.
Observation 2. and for each .
For all values of , the following result provides the -metric antidimension of Jahangir graphs.
Theorem 2. For , let be a Jahangir graph. Then,
Proof. First of all, it is worthy to note that and , for any , and , for any . Now, we need to discuss the following seven claims. Claim 1: the set is an -antimetric generator for . Note that , for all , and , for all . According to the relation , there are only two equivalence classes each has cardinality . So, the result followed by Remark 1. Claim 2: every singleton subset of is a 3-antimetric generator for . Let for any fixed . Then, , for all , , for all , and , for all . Hence, the relation supplies three equivalence classes , , and . Thus, , and hence, is 3-antimetric generator, by Remark 1. Claim 3: every singleton subset of is a 2-antimetric generator for . Let ; then, metric codes of the vertices are Clearly, we receive four equivalence classes according to the relation , , , , and . Hence, , and is a antimetric generator, by Remark 1. Claim 4: every element subset of is either 1-antimetric generator or 2-antimetric generator for . Let be a 2-element subset of . Then, we have the following two cases to discuss. Case 1 ( contains ): here, we have two subcases. Subcase 1.1: let with ; then, , for distinct : So, the equivalence classes corresponds to the relation are , , and . Here, , which implies that is a 2-antimetric generator, by Remark 1. Subcase 1.2: let with ; then, , for : Due to the above metric coding, it is clear that we find four equivalence classes according to the relation , which are , , , and . Thus, , which implies that is a antimetric generator, by Remark 1. Case 2 ( does not contain ): again, we have three subcases to discuss. Subcase 2.1: let and . Then, . If , then a vertex , such that , has the unique metric code from the set with respect to . If no vertex from is a common neighbor of and , then the vertex has the unique metric code with respect to . Subcase 2.2: let and . Then, either or . In the former case, a vertex , such that , has the unique metric code . In the later case, we have two possibilities. If there is a vertex such that , then a vertex , with and , has the unique metric code with respect to . If there is no such in , then the vertex has the unique metric code with respect to . Subcase 2.3: let for and . Then, either or . For the later case, let , where are distinct vertices. Here, we have two possibilities. If one of the neighbors and of , say , has the property that , then the neighbor of has the unique metric code with respect to . If , then the vertex has the unique metric code with respect to . In the former case, is a one neighbor of from , and the other neighbor of from has the unique metric code with respect to . In each possibility of these subcases, the relation proposes at least one singleton equivalence class, which follows that . Hence, is an antimetric generator, by Remark 1. Claim 5: for , the set is a antimetric generator whenever . Otherwise, is antimetric generator for . If , the following two cases are need to be discussed for . Case 1: whenever , metric codes with respect to are So, the equivalence classes corresponds to the relation are , , , , and . Note that , so Remark 1 yields the required result. Case 2: whenever , metric codes with respect to are Hence, the equivalence classes corresponds to relation are , , , and . It follows that because . Thus, Remark 1 proves that is a 2-antimetric generator. Now, if , then again we have two cases. Case 1: whenever , we have a vertex such that , for all Case 2: whenever , we have a vertex with and , for all In both the cases, we get at least one singleton equivalence class with respect to the relation , which implies that . Hence, is a antimetric generator, by Remark 1. Claim 6: except , the set is a 2-antimetric generator whenever . Otherwise, is a antimetric generator for . Whenever , then the metric coding with respect to is Hence, we have the equivalence classes , , , , , and in accordance with the relation . Thus, , o is a 2-antimetric generator, by Remark 1. Next, whenever , then In each possibility, the given metric code is unique, which provides a singleton equivalence class according to the relation . Hence, , and is a antimetric generator, by Remark 1. Claim 7: every set of cardinality is a antimetric generator for , except the sets and discussed in Claims 5 and 6, respectively.If contains the vertex , then there exists a vertex (or ) such that is a neighbor of some element in , and , for any . If does not contain the vertex , then has the unique metric code with respect to . In both the cases, we get a singleton equivalence class according to the relation . Hence, , and Remark 1 implies that is a 1-antimetric generator.
All these claims conclude the proof with the following points:(i)For , there does not exist a -antimetric generator for .(ii)Claims , and 3 provide that . Furthermore, there exist 1-antimetric generators for of cardinality at least 2, by Claims 4 to 7. It follows that .(iii)Claim 3 provides the existence of a 2-antimetric generator for of cardinality 1, which yields that .(iv)Claim 2 provides the existence of a 3-antimetric generator for of cardinality 1, which implies that .(v)An -antimetric generator for of cardinality 1 exists due to Claim 1, and hence, .
3.2. Helm Graphs
For , a helm graph, , is obtained from a wheel graph by attaching one leaf (a vertex of degree one) with each vertex of the cycle . Let be the set of leaves; then, the vertex set of a helm graph is , and its edge set is , where the indices greater than or less than 1 will be taken modulo . One helm graph is shown in Figure 3.
It is an easy task to verify the following observation.
Observation 3. When , , for . While
Theorem 3. For , let be a helm graph. Then,
Proof. It is significant to keep in hand the neighborhoods and , for any , and , for any leaf . Now, we have to discuss the following eight claims. Claim 1: the set is an antimetric generator for . Note that , for all , and , for all . There are only two equivalence classes and , both of cardinality , according to the relation . Hence, , which implies that is an -antimetric generator, by Remark 1. Claim 2: every single leaf form a 1-antimetric generator for . Let be a set of one leaf . Then, , where , and no other vertex of has the code similar to . It follows that there exists a singleton equivalence class due to the relation . Accordingly, Remark 1 refers that is an antimetric generator. Claim 3: every singleton subset of is a 4-antimetric generator for . Let , for any fixed ; then, , for all , , for all , and , for all . Thus, the relation produces three equivalence classes , , and . Hence, , and is a 4-antimetric generator for , by Remark 1. Claim 4: the set is a 3-antimetric generator for whenever . Otherwise, is a 1-antimetric generator. Whenever , we have , and the metric codes with respect to are The equivalence classes corresponds to the relation are , , and . It follows, by Remark 1, that is a 3-antimetric generator since . When , then the vertex has the metric code which is not similar to the metric code of any other vertex of . This creates at least one singleton equivalence class in accordance with the relation , which implies that is a 1-antimetric generator for , by Remark 1. Claim 5: every element subset of , except the set of Claim 4, is a 1-antimetric generator for . Let be a element subset of and . Then, we discuss the following two cases: Case 1 ( does not contain ): let . When , the vertex has the unique metric code with respect to . When , the vertex has the unique metric code with respect to . Otherwise, the vertex has the unique metric code with respect to . Next, we let . When , the vertex has the unique metric code with respect to . Otherwise, the vertex has the unique metric code with respect to . Case 2 ( contains ): let ; then, the leaf has the unique metric code with respect to . Each possibility in both the cases yields at least one singleton equivalence class according to the relation , which implies that is a 1-antimetric generator, by Remark 1. Claim 6: the set is a 2-antimetric generator for . Note that , , So, there are four equivalence classes , , , and with respect to the relation . That is, , and Remark 1 assists that is a 2-antimetric generator. Claim 7: for even values of , the set is a 2-antimetric generator for . Note the metric coding of the vertices with respect to is as follows: Hence, the classes according to the relation are , , , , , and . Here, , which implies that is a 2-antimetric generator, by Remark 1. Claim 8: any subset of of cardinality is a 1-antimetric generator for , except the sets and considered in Claims 6 and 7, respectively.Let be a subset of with and . Then, note the following two possibilities:(1) contains : if and with , but (because this case is already discussed in Claim 6). Then, the vertex has the unique metric code from the set with respect to . If , then a neighbor of some receives the unique metric code with respect to .(2) does not contain : the vertex has the unique metric code with respect to .In both the possibilities, we get at least one singleton equivalence class according to the relation . Thus, , and Remark 1 provides that is a 1-antimetric generator.
The proof will reach to its end by discussing the following points on the base of formerly discussed claims:(i)For , there does not exist a -antimetric generator for .(ii)We find a 1-antimetric generator for of (1) cardinality 1 due to Claim 2, (2) cardinality 2 due to Claims 4 and 5, and (3) cardinality due to Claim 8. Since a 1-antimetric generator for of cardinality 1 is the smallest one, so .(iii)Claim 6 assures the existence of a 2-antimetric generator for of cardinality 3 for all values of , while Claim 7 assures the existence of a 2-antimetric generator for of cardinality 5 just for even values of . Moreover, no singleton set or 2-element set of vertices in is a 2-antimetric generator for due to Claims 1 to 5. It follows that .(iv)We receive a antimetric generator of cardinality 2 from Claim 4, and no singleton set is a 3-antimetric generator for due to Claims 1 to 3, which implies that .(v)Claim 3 assists that because of the existence of a 4-antimetric generator for of cardinality 1.(vi)Finally, due to an -antimetric generator for of cardinality 1 exists by Claim 1.
3.3. Flower Graphs
For , a flower graph, , is obtained from a helm graph by joining its each leaf to the vertex of . Accordingly, the vertex set of a flower graph is , and its edge set is , where the indices greater than or less than 1 will be taken modulo . Figure 4 provides graphical appearance of one flower graph.
The following observation is easy to understand for the flower graph .
Observation 4.
Theorem 4. For all , let be a flower graph. Then,
Proof. The following listed neighborhoods of the vertices of will be used in the proof: and , for any , and , for any . We have to discuss the following nine claims to prove the required result. Claim 1: the set is a -antimetric generator for . Note that , for all . So, the only one equivalence class of cardinality is produced by the relation . Hence, is a -antimetric generator. Claim 2: every singleton subset of is a antimetric generator for . Let for any fixed ; then, , for all , and , for all . The relation creates two equivalence classes and . It follows that , and Remark 1 proposes that is a 2-antimetric generator. Claim 3: every singleton subset of is a 4-antimetric generator for . Let for any fixed ; then, , for all , and , for all . Thus, we get two equivalence classes and by the relation with . Hence, is a 4-antimetric generator, by Remark 1. Claim 4: the set is a 3-antimetric generator for . Note the metric codes with respect to is as follows: , for all , and , for all . Here, the equivalence classes obtained through the relation are and . Hence, , and Remark 1 implies that is a 3-antimetric generator. Claim 5: the set is a 2-antimetric generator for , where . The metric codes with respect are , for all , , for all , and , for all . Accordingly, three equivalence classes , , and are generated by the relation with . Therefore, Remark 1 provides that is a antimetric generator. Claim 6: any 2-element set, , is a 1-antimetric generator for , except the sets and discussed in Claims 4 and 5, respectively. We discuss the following two possibilities:(1)Either or or with . Then, is the unique metric code in .(2)If , then is the unique metric code in . In both the possibilities, we receive at least one singleton equivalence class, in accordance with the relation , which implies that . Hence, Remark 1 yields that is a 1-antimetric generator. Claim 7: the set is a 2-antimetric generator for . The metric coding with respect to is It follows that and are the equivalence classes produced by the relation . Here, , which implies that is a 2-antimetric generator, by Remark 1. Claim 8: the set is a 3-antimetric generator for whenever . Otherwise, is a antimetric generator for . , and we have the metric codes of the vertices with respect to as follows: Thus, the relation partitioned into three equivalence classes , , and , with . It follows, by Remark 1, that is a 3-antimetric generator. Whenever , if or , then a neighbor of lying in has the unique metric code with respect to . If or , then or , respectively, has the unique metric code with respect to . In either cases, we obtain at least one singleton equivalence class according to the relation , which implies that is a 1-antimetric generator, by Remark 1. Claim 9: any set of cardinality is a antimetric generator for , except the sets and discussed in Claims 7 and 8, respectively. We discuss the following two cases: Case 1 ( does not contain ): in this case, the vertex has the unique metric code with respect to Case 2 ( contains ): since , there exists a vertex for at least one such that has the unique metric code with respect to In both the cases, we get at least one singleton equivalence class according to the relation , which implies that is a antimetric generator, by Remark 1.
We conclude the proof by discussing the following points using preceding claims:(i)For , there does not exist a -antimetric generator for .(ii)We get an -antimetric generator for of (1) cardinality 2 by Claim 6 and (2) cardinality by Claims 8 and 9. Furthermore, no singleton set possesses the property of antimetric generator in , by Claims 1 to 3. It follows that .(iii)For , 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 .(iv)There exists a 3-antimetric generator for of cardinality 2 due to Claim 4, and a 3-antimetric generator of cardinality 3 due to Claim 8. Thus, Claims 1 to 3 conclude that .(v)Claim 3 declares the existence of a 4-antimetric generator for of cardinality 1, which implies that .(vi)A -antimetric generator for exists due to Claim 1, so .
3.4. Sunflower Graphs
For , a sunflower graph, , is obtained from a wheel graph by attaching one vertex to every two consecutive vertices of the cycle . Let ; then, the vertex set of a sun flower graph is and its edge set is , where the indices greater than or less than 1 will be taken modulo . A graphical preview of this graph is displayed in Figure 5.
The following observation is an easy exercise to understand.
Observation 5. When , , for . While
Theorem 5. For , let be a sunflower graph. Then,
Proof. The neighborhoods, and for any , and , for any , of the vertices in are useful to discuss the following nine claims. Claim 1: the set is an -antimetric generator for . Note that , for all , and , for all . Thus, there are two equivalence classes and according to the relation , and each class has elements. Hence, Remark 1 yields that is an -antimetric generator. Claim 2: every singleton subset of is a antimetric generator for . Let for any fixed . Then, , for all : Therefore, the equivalence classes, corresponding to the relation , are , , and . It follows that , and is a 5-antimetric generator, by Remark 1. Claim 3: every singleton subset of is a antimetric generator for . Let , for any fixed . Then, We have four equivalence classes , , , and , in accordance with the relation . Thus, , which implies that is a 2-antimetric generator, by Remark 1. Claim 4: the set is a 2-antimetric generator for . The metric coding with respect to is listed as follows: So, the relation supplies five equivalence classes , , , , and . Hence, , and is a 2-antimetric generator, by Remark 1. Claim 5: the set is a 2-antimetric generator for . We have the following metric coding with respect to : , for all : , for all , and , for all . So, the equivalence classes, in accordance with the relation , are , , , , , and . Here, , which implies that is a 2-antimetric generator, by Remark 1. Claim 6: each 2-element set is a antimetric generator for . We discuss the following three possibilities:(1)Let for and . If either or , then there is a vertex in such that , and , for any . If , then there is a neighbor of from such that , for any . If , then the vertex has the unique metric code with respect to .(2)Let and . Then, either or . In the former case, a vertex , for which and , has the unique metric code with respect to . In the later case, we have two discussions: If , then a vertex , such that , has the unique metric code from the set with respect to . If no vertex in is a common neighbor of and , then the vertex has the unique metric code with respect to .(3)Let ; then, is the unique metric code in . In all these possibilities, we get at least one singleton equivalence class according to the relation , which implies that . Hence, is a 1-antimetric generator, by Remark 1. Claim 7: the set is a 2-antimetric generator of whenever . Otherwise, is a 1-antimetric generator. Whenever , note that and Hence, we have eight equivalence classes , , , , , , , and in accordance with the relation . It can be seen that , which yields that is a 2-antimetric generator, by Remark 1. Whenever , we have a vertex such that for any . Hence, we receive at least one singleton equivalence class due to the relation , which implies that is a 1-antimetric generator, by Remark 1. Claim 8: the set is a 2-antimetric generator for whenever . Otherwise, is a 1-antimetric generator. Whenever , we have the metric codes with respect to as follows: Therefore, we get 10 equivalence classes , , , , , , , , , and in accordance with the relation . It has been observed that , so is a antimetric generator, by Remark 1. Whenever , we have a vertex such that , for any . Hence, we receive at least one singleton equivalence class by the relation , which implies that . Hence, is a antimetric generator, by Remark 1. Claim 9: except the sets discussed in Claims 7 and 8, respectively, each set of cardinality is a antimetric generator for . We have to discuss the following two cases: Case 1 ( contains ): let (because the case, when , has been discussed in Claims 7 and 8). Then, there a vertex such that is a neighbor of some whenever , or there a vertex such that is a neighbor of some whenever either or , and we get the unique metric code of with respect to . Case 2 ( does not contain ): whenever or , the vertex has the unique metric code of with respect to . Whenever , there is a vertex (or ) such that for some element , and , for any .In both the cases, the relation supplies at least one singleton equivalence class, which yields that . Hence, is a 1-antimetric generator, by Remark 1.
These claims complete the proof with the following deductions:(i)There does not exist a antimetric generator for when .(ii)Claim 6 supplies a antimetric generator for of cardinality 2, and Claims 7 to 9 supply a antimetric generator for of cardinality . Claims 1 to 3 provide the guaranty of nonexistence of singleton antimetric generator for . It follows that .(iii)The existence of a antimetric generator for of cardinalities 1, 2, and 3 is assured by Claim 3, by Claims 4 and 5, and by Claims 7 and 8, respectively. Accordingly, .(iv) because of the existence of a antimetric generator for of cardinality 1 in Claim 2.(v)Claim 1 provides an antimetric generator for of cardinality 1, which yields that .
4. Concluding Remarks
For a connected graph , the number is called the radius of , where is the eccentricity of . The center of is a subgraph induced by the set . It has been observed the following useful properties about a antimetric dimensional graph in [7].
Remarks 2. (see [7])(a)If a connected graph is metric antidimensional, then (b)If the center of a connected graph is trivial, then it is metric antidimensional for some
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 metric antidimension, for some , and it must be metric antidimensional for some . So, naturally, it raises the following two questions: Q1. For how many and for which values of a wheel-related social graph admits metric antidimension? Q2. For which maximum value of , , a wheel-related social graph is metric antidimensional?
The results of Čangalović et al., proved in [1] and listed in Observation 1 and Theorem 1, were the pioneers to address the answers of questions Q1 and Q2. These results revealed that (1) when , a wheel graph admits metric antidimension for three values of and (2) when and for all , admits metric antidimension for four values of . It follows that is metric antidimensional.
To extend the study of anonymity based on the metric antidimension, we considered four graphs related to wheel graphs in this article. By investigating their metric antidimension, we addressed the answers of questions Q1 and Q2 as follows:(i)For a Jahangir graph , Observation 2 and Theorem 2 revealed that (1) admits metric antidimension for three values of , (2) when , admits metric antidimension for three values of , and (3) for all , admits metric antidimension for four values of (ii)For a helm graph , Observation 3 and Theorem 3 revealed that (1) when , admits metric antidimension for three values of , (2) admits metric antidimension for four values of , and (3) for all , admits metric antidimension for five values of (iii)For a flower graph , Observation 4 and Theorem 4 revealed that (1) admits metric antidimension for three values of and (2) for all , admits metric antidimension for five values of .(iv)For a sunflower graph , Observation 5 and Theorem 5 revealed that (1) when , admits metric antidimension for three values of , (2) when , admits metric antidimension for four values of , (3) admits metric antidimension for four values of , and (4) for all , admits metric antidimension for four values of .
From all these results, it can be concluded that each considered wheel-related social graph is metric antidimensional.
Furthermore, according to the computed metric antidimension of wheel-related social graphs, we investigated that each of them meets the anonymity in the following ways (skipping particular cases which can be observed straightforwardly). Wheel graph: for and and by Theorem 1, we have Jahangir graph: for and and by Theorem 2, we have Helm graph: for and and by Theorem 3, we have Flower graph: for and and by Theorem 4, we have Sunflower graph: for and and by Theorem 5, we have
The anonymity, measured on the base of metric antidimension for the maximum value of , assures that a user can be reidentified with the probability less than or equal 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.
Conjecture 1. Each wheel-related social graph and generalizations of wheels are metric antidimensional and meet anonymity.
Data Availability
The figures, tables, and other data used to support this study are included within the article.
Conflicts of Interest
The authors declare that there are no conflicts of interests regarding the publication of this paper.
Acknowledgments
This work was supported in part by general project of Anhui University excellent talent support plan 2021 under Grant Number gxyq2021235.