Abstract
Covering, matching, and domination are the basic concepts in graphs that play a decisive role in the properties of graphs. Calculating these parameters is one of the difficulties in fuzzy graphs when it is not possible to accurately determine the values of the vertices of a graph. The intervalvalued intuitionistic fuzzy graph (IVIFG) is one of the fuzzy graphs which can play an important role in solving uncertain problems in different sciences such as psychology, biological sciences, medicine, and social networks. The necessity of using a range of value instead of one number caused them to help researchers in optimizing and saving time and cost. In this study, we introduce some of the specific concepts such as covering, matching, and paired domination using strong arc or effective edges by giving appropriate examples. In addition, we have calculated strong node covering number, strong independent number, and other parameters of complete bipartite IVIFGs with several examples. Finally, we have presented an application of IVIFG in social networks.
1. Introduction
Graphs are an inevitable tool in applied mathematics. Among the various concepts in graphs, some concepts are more important such as covering, matching, and domination. These concepts are closely related to vertices, as one of the most important components of the graph, and cause them to participate in many analyses related to vertices. Many studies have been done by researchers on various graphs. It is difficult to examine these concepts when the exact values for the vertices cannot be considered.
In 1965, Zadeh [1] presented the basic idea of fuzzy set (FS) where its prominent feature was the allocation of membership degree between 0 and 1 to each element in a set. Zadeh [2] also introduced the intervalvalued fuzzy set (IVFS) in 1975, in which membership degrees were intervals of numbers. Roselfeld [3] defined a new concept called the fuzzy graph (FG) by employing fuzzy relations on FS. FGs were considered by researchers in the fields related to ambiguous and uncertain problems. They were able to find numerous applications in solving and modeling problems in computer science, engineering, system analysis, economics, network routing, transportation, and so on. With the advent of new indefinite problems, it became clear that a membership function could not well express the ambiguity in subjective perceptions and the complexity of data. To overcome this shortcoming of the FS, Atanassov [4] proposed an extension of FS by introducing nonmembership function and defined intuitionistic fuzzy set (IFS). IFGs were first introduced by Atanassov [5] in 1999 and was further discussed in [6]. Mahapatra et al. [7–9] explored concepts on fuzzy graphs. Rashmanlou and Pal [10, 11] studied different kinds of FGs. Kosari et al. [12] presented vague graph structure with an application in medical diagnosis. Kou et al. [13] studied some properties of vague graph (VG). Krishna et al. [14] studied new results in cubic graphs. Talebi et al. [15, 16] defined CayleyFGs and some operations on level graphs of bipolar FGs. Atanassov [17] recently introduced some new topological operators over IFSs. Mathew et al. [18] conducted research on vertex rough graphs. Some concepts in IVFGs and neutrosophic graphs are studied by Jan et al. [19]. Voskoglou [20] used a combination of soft sets and gray numbers in decisionmaking. Mahapatra et al. [21–23] introduced concepts of neutrosophic graphs used in social networks.
The decision to determine accurate numerical values in uncertain and inaccurate evaluations of information, which often occur in practical situations, is associated with difficulties. Thus, in 1989, Atanassov and Gargov [24] introduced the idea of the intervalvalued intuitionistic fuzzy set (IVIFS) in order to unify perceptions and quantify the uncertain nature of the mind. This concept is defined by a membership function, a nonmembership function, and a hesitant function whose values are intervals between 0 and 1 instead of exact numbers. IVIFS has been widely used in many areas, such as decisionmaking [25], pattern recognition [26], medical diagnosis [27], and graph theory [28]. The concept of intervalvalued fuzzy graphs (IVFGs) is presented by Hongmei and Lianhua in [29]. Akram et al. [30, 31] defined certain types of IVFGs. The product of IVIFGs was proposed by Mishra and Pal in [32]. The strong IVIFG concept is described by Ismayil and Ali [33]. Rashmanlou et al. [25, 34–36] studied some IVIFG concepts.
The purpose of this paper is to find a way to determine the concepts of vertex covering, matching, and paired domination in IVIFGs where we are dealing with intervalvalued numbers instead of fuzzy numbers. The previous definition limitations in the vertex covering, matching, and paired domination of FGs have directed us to offer new classifications in terms of IVIFG. These concepts have already been studied by some researchers in a variety of FGs. Sahoo et al. [37] investigated covering and paired domination in IFGs.
The rest of this article is organized as follows: Section 2 briefly reviews related basic concepts to IVIFGs. In Section 3, we introduced the concepts of strong vertex covering, independent vertex covering, and perfect strong matching in an IVIFG by strong edges and defined some of its properties in specific types of IVIFGs. In this section, we introduce paired domination in IVIFG and examine its implications. Finally, we present an application of IVIFG on social networks in Section 4.
2. Preliminaries
In this section, we briefly define some of the basic concepts for entering the main discussion.
Definition 1 (see [24]). An IVIFS in can be described as
where , , and , for all .
Similarly, the intervals and denoted the MV and nonMV of an element , respectively. If each of the intervals contains only one value for each , we have
Furthermore, the hesitancy degree of each element is as follows:
Definition 2 (see [33]). An IVIFG of an underlying graph is a pair so that is an IVIFS in and is an intervalvalued intuitionistic fuzzy relation (IVIFR) so that and , for each .
Definition 3 (see [34]). An edge of an IVIFG, is named a strong arc (SA) or effective edge if
Definition 4 (see [34]). An IVIFG is complete, if for all .
As a result of the above definition, the following definition can be provided.
Definition 5. An IVIFG is named bipartite whenever the vertex set can be partitioned into two nonempty sets and so that and , for or . If for all and ; then, is named a complete bipartite IVIFG (CBIVIFG) and is shown by .
All the basic notations are shown in Table 1.
3. Covering, Matching, and Paired Domination in the IVIFGs
In this section, we introduce covering, matching, and paired domination in the IVIFGs by the weight of strong edges and examine some of its properties and results.
Definition 6. Let be an IVIFG. An SNC in an IVIFG is the set of nodes that cover all SAs of . The weight of an SNC is denoted as
so that and are the minimum of the lower and upper of IVMBs and and are the maximum of the lower and upper of IVNMBs of all SAs incident on , respectively.
An SNCN of an IVIFG is shown as follows so that
A minimum SNC in an IVIFG is an SNC of minimum IVMBs and maximum IVNMBs.
Theorem 7. Let be a CIVIFG. Then, where and are the lower and upper of IVMBs and and are the lower and upper of IVNMBs of the weakest arc in . Note that is the number of vertex in .
Proof. Since is a CIVIFG, all arcs are strong, and every node is neighbor to all other vertices. So, any set includes nodes forming an SNC of .
Let be a vertex having minimum of IVMBs and maximum of IVNMBs in . Suppose is the node neighbor to . Then, the arcs are all weakest arcs of , and strength of each arcs is equal to which .
Hence, the set of vertices forms an SNC of with
where is the minimum lower of IVMB and is the minimum upper of IVNMB of SAs incident on . Then,
where and are the lower and upper of IVMBs of a weakest arc in .
Hence, and . Similarly,
where and , are the maximum lower and upper of IVNMBs of all SAs incident on . Then,
where and are the lower and upper of IVNMBs of a weakest arc in . Hence, and .
Theorem 8. For a CBIVIFG with partite set and ,
Proof. All arcs in are strong, and each node in is neighbor with all nodes in and contrariwise. The set of all arcs of is a set of all arcs incident on each node of or a set of all arcs incident on each node of . Hence, all SNCs in are , and . Clearly, is greater than and . Hence, Similarly, . Also, is less than and . So, In the same way, we have .
Definition 9. In an IVIFG , two nodes are said to be SI if there is no SA between them. A set of nodes in is an SI if and only if two nodes are in an SI set.
Definition 10. The weight of an SIS in an IVIFG is described as
i.e.,
where and are minimum of the lower and upper of IVMBs and and are maximum of the lower and upper of IVNMBs of all SAs incident on , respectively.
An SIN of an IVIFG is shown by , which
A maximum SIS in an IVIFG is an SIS with the maximum IVMBs and minimum IVNMBs.
Theorem 11. Let be a CIVIFG. Then, where and are the lower and upper of IVMBs and and are the lower and upper of IVNMBs of a weakest arc in .
Proof. Since is a CIVIFG, so all arcs are strong, and also, each arc is neighbor to all other nodes. Hence, is the only SIS for each . Thus, the result is true.
Theorem 12. Let be a CBIVIFG with partite set and . Then,
Proof. In all arcs are strong. Also, each node in is neighbor with all nodes in and contrariwise. Therefore, all SISs in are and . Hence, the result is true.
Example 1. Consider an IVIFG is drawn in Figure 1.
Clearly, all arcs are strong, and all SNCs of are as follows:
Table 2 shows the method of calculating the weight of SISs.
Thus, .
Example 2. Consider a strong IVIFG is drawn in Figure 2. All SISs in are . The calculation of the weight of SIS is shown in Table 3. Therefore, .
Definition 13. Let be an IVIFG without INs. The weight of an SAC is described as , which .
An SACN of an IVIFG is denoted by , where
A minimum SAC in an IVIFG is an SAC with minimum IVMBs and maximum IVNMBs.
Theorem 14. If is a complete IVIFG, then
Proof. Since is a CIVIFG, so all arcs are SA, and each vertex is neighbor to all others vertices. Also, the number of arcs in SAC of both and is identical because each arc in both graphs is strong. Now, the SACN of is . Therefore, the minimum number of arcs in an SAC of is . This completes the proof.
Theorem 15. If is a CBIVIFG with partite set and . Then,
Proof. In , all arcs are strong. Also, each node in is neighbor with all nodes in and contrariwise. Also, the number of arcs in an SAC of both and is identical because each arc in both graph is strong. Now, the arc covering number of is . Therefore, the minimum number of arcs in an SAC of is . Thus, the result is obtained.
Definition 16. Let be an IVIFG. A set of SAs in so that no two arcs in have a common node is named an SIS of arcs or an SM in .
Definition 17. Let be an SM in IVIFG . If , then, we say that strongly matches to . The weight of an SM is described as An SMN of an IVIFG is shown by , which A maximum SM in an IVIFG is an SM of maximum IVMBs and minimum IVNMBs.
Theorem 18. If is a CIVIFG, then
Proof. Since is a CIVIFG, all arcs are strong, and each node is neighbor to all other nodes. Also, the number of arcs in an SM of both and is identical because each arc in both graph is strong. Now, the SMN of is . Therefore, the maximum number of arcs in an SM of is . Hence, the result follows.
Theorem 19. For a CBIVIFG with partite set and ,
Proof. In , all arcs are strong. Also, each node in is neighbor with all nodes in and contrariwise. Thus, the number of arcs in an SM of both and is identical because each arc in both graphs is strong. Now, the SMN of is . Therefore, the maximum number of arcs in an SM of is .
Hence, the result is obtained.
Example 3. Consider a strong IVIFG is drawn in Figure 3. All arcs are strong, and the SACs are as follows:
The calculation of the weight of SACs is shown in Table 4.
So, .
Again, the two sets and are the only SAC and SM in . So,
Hence, .
Example 4. Consider an IVIFG is drawn in Figure 4.
All SAs are , , and , and all SACs are as follows:
The calculation of the weight of SACs is shown in Table 5. So, .
The set is the only SIAC. So, .
Theorem 20. Let be an IVIFG containing no IN. Then,
Proof. Let be a minimum SNC of , which Then, is an SIS of nodes. In other words, the nodes in are incident on SAs of . Thus, Let , where is a maximum SIS of nodes in . That is, no two nodes in are neighbor to each other by an SA, and thus, the node in strongly covers all SAs of . Hence, is an SNC of , and and are the minimum lower and upper of IVMBs, and and are the maximum lower and upper of IVNMBs. So, From (38) and (39), we have
Definition 21. Let be an IVIFG and be an SM in . Then, is named a PSM if strongly matches each node of to some nodes of .
Example 5. Consider an IVIFG is drawn in Figure 5. All arcs are strong, and the sets and are PSMs. The calculation of the weight of PSMs is given in Table 6.
So, .
Hence, .
Now, we introduced PD in IVIFGs using SAs based on PSM. Also, some useful results are established.
Definition 22. A set of nodes of IVIFG is an SDS of if every node of is a strong neighbor of some nodes in .
Definition 23. The weight of an SDS is defined as
or
where and are the minimum lower and upper of IVMBs and and are the maximum lower and upper of IVNMBs of SAs incident on , respectively.
An SDN of an IVIFG is denoted by , where
Definition 24. Let be an IVIFG. A set of nodes is named to be an SPDS if is an SDS and the IVIFsubgraph induced by has a PSM. The weight of an SPDS is described as , which An SPDN of an IVIFG is denoted by , that
Example 6. Consider an IVIFG is drawn in Figure 6. All SAs are , and . The PDs in are , , and . The weights of these sets are calculated as follows:
Table 7 shows the calculation of the weight of PDs.
Hence, .
Theorem 25. Let be a CIVIFG. Then, where and are the lower and upper of IVMB and and are the lower and upper of IVNMB of any weakest arc in , respectively.
Proof. Since is a CIVIFG, all arcs are strong, and every vertex is neighbor to all other vertices. Then, any set consisting of two nodes in forms an SPDS. Hence, where is the weakest arc in .
4. Application
Social networks are a group of individuals or organizations with common tastes or interests that come together to achieve specific goals. Each member is named an actor. Social networks are characterized by complex relationships and interactions between actors. The main reasons for creating social networks are individual relationships, labor relations, scientific relations, shared tastes, interests and hobbies, sociopolitical motives, and virtual network analysis.
Graphs are used as a mathematical tool to represent and analyze a social network by visually representing social networks. In these graphs, the actors are considered as vertices of the graph, and the connections between them are displayed by the edges of the graph. Intuitively, the edges are distributed on social networks locally. This means that the number of edges distributed among a group of vertices is much greater than the number of distribution edges among this group of vertices and the rest of the vertices of the graph. This feature, which can be seen in graphs related to real data, is called a community. In some sources, the community is also called a cluster or module. In other words, communities are a set of vertices that are more likely to share common features than the rest of the graph. Since people in forums on a social network are more likely to have common interests, this information can be used to promote specific products by finding their interests. Most online social networks have overlapping communities. This means that these networks are made up of overlapping communities, and one vertex can belong to more than one community. Figure 7 illustrates the social network of researchers in a country that is a member of different scientific communities according to the subjects under study. These communities include chemistry, biology, engineering, information technology (IT), mathematics, medicine, physics, and social sciences. Table 8 shows the number of members of each community and the average number of members present at the meetings.
In the evaluations made by the members on the effect of community on the scientific promotion of members, since the mentioned variables have uncertain values, so for each community, we considered an intervalvalued intuitionistic fuzzy number as the amount of influence of community on its members. Since the presence of members in the meetings of the community is effective on the scientific promotion of members, we introduced the ratio of the average number of members present in the meetings to the total number as an IVIFN. For example, studies have shown that the biology community is 80 to 90 percent effective in advancing the science of its members and 5 to 10 percent ineffective. These values are specified in Table 9.
The strong relationships between scientific communities are illustrated in the form of an IVIFG in Figure 8. In this IVIFG, the membership values of the edges are the effect that the members of the two communities have on their scientific advancement. For example, the collaboration between the two communities of chemistry and medicine is about 79 to 89 percent effective in the scientific advancement of the members of each community and 11 to 21 percent ineffective. These values are shown in Table 10.
In general, there is no polynomial algorithm for finding a maximum independent set for an arbitrary graph. This means that it is not possible to access such a collection in a short time. To obtain the maximal SISs in IVIFG with a small number of vertices, we use the following instructions.

Since all edges are SA, so by applying the above steps for all vertices on the IVIFG of Figure 8, all maximal SISs and cardinalities can be seen in Table 11. Now, by calculating the cardinal of all the SISs obtained from the above steps, we can also determine the maximum SISs.
The maximum SISs are , , and .
After calculating the weight of the above sets, we have , , and .
Therefore, has the maximum weight of membership and the minimum weight of nonmembership, so it can be chosen as the best option. It is interesting to know that also has the maximum number of members in scientific communities. That is, strong independent scientific communities include chemistry, engineering, and social sciences.
Suppose knowledgebased companies intend to organize an exhibition at the meeting place of scientific communities to acquaint researchers with their scientific products. Researchers at the knowledgebased companies can be members of various scientific communities. The goal is to hold as many exhibitions as possible at the same time provided that each knowledgebased company has a maximum of one exhibition in a specific time period and to hold another exhibition at different time intervals. In this case, the maximum independent set is the maximum number of exhibitions that can be held at one time in scientific communities.
5. Conclusion
Analysis of uncertain problems by IVIFG is important because it gives more integrity and flexibility to the system. An IVIFG, as an extension of FGs, has good capabilities in dealing with problems that cannot be explained by FGs. They have been able to have wide applications even in fields such as psychology and identifying people based on cancerous behaviors. In this paper, covering and matching have been defined in IVIFGs using strong arcs. These concepts are introduced as an intervalvalued intuitionistic number. One of the advantages of this method is that the amount of defined parameters can be expressed and compared in terms of membership and nonmembership. Also, the concepts of SNC, SIN, SAC, and SM in IVIFGs are determined, and the relations among them have been obtained. Furthermore, we have introduced the PD and SPDN in CIVIFG and CBIVIFG. Since the parameters being studied are interval values, comparisons of these parameters may be limited in an IVIFG. Finally, we have presented an application of IVIFG in social networks. In their future work, the authors try to study the concepts of polar IVIFGs.
Data Availability
No data were used in this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the Natural Science Foundation of Guangdong Province of China (2022A1515011468).