Abstract
Fuzzy graph (FG) models take on the presence being ubiquitous in environmental and fabricated structures by human, specifically the vibrant processes in physical, biological, and social systems. Owing to the unpredictable and indiscriminate data which are intrinsic in real-life, problems being often ambiguous, so it is very challenging for an expert to exemplify those problems through applying an FG. Vague graph structure (VGS), belonging to FGs family, has good capabilities when facing with problems that cannot be expressed by FGs. VGSs have a wide range of applications in the field of psychological sciences as well as the identification of individuals based on oncological behaviors. Therefore, in this paper, we apply the concept of vague sets (VSs) to GS. We define certain notions, VGS, strong vague graph structure (SVGS), and vague -cycle and describe these notions by several examples. Likewise, we introduce -complement, self-complement (SC), strong self-complement (SSC), and totally strong self-complement (TSSC) in VGS and investigate some of their properties. Finally, an application of VGS is presented.
1. Introduction
The FG concept serves as one of the most dominant and extensively employed tools for multiple real-word problem representations, modeling, and analysis. To specify the objects and the relations between them, the graph vertices or nodes and edges or arcs are applied, respectively. Graphs have long been used to describe objects and the relationships between them. Many of the issues and phenomena around us are associated with complexities and ambiguities that make it difficult to express certainty. These difficulties were alleviated by the introduction of fuzzy sets by Zadeh [1]. A GS, proposed by Sampathkumar [2], refers to the generalization of undirected graph being relatively beneficial in investigating some structure including graphs, signed graphs, and graphs in which every edge is labeled or colored. A GS facilitates studying the different relations and the equivalent edges simultaneously. The FS focuses on the membership degree of an object in a particular set. The existence of a single degree for a true membership could not resolve the ambiguity on uncertain issues, so the need for a degree of membership was felt. Afterward, to overcome the existing ambiguities, Gau and Buehrer [3] introduced false-membership degrees and defined a VS as the sum of degrees not greater than 1. Kaufmann [4] represented FGs based on Zadeh’s fuzzy relation [5, 6]. Rosenfeld [7] described the structure of FGs obtaining analogs of several graph theoretical concepts. Harinath and Lavanya [8] studied new concepts in fuzzy graph structures. Bhattacharya [9] gave some remarks on FGs. Several concepts on FGs were introduced by Mordeson and Nair [10]. Dinesh [11] investigated the notion of a FGS and studied some related properties. Ghorai et al. [12, 13] presented fuzzy tolerance graphs and planarity in VGs. Ramakrishna [14] defined VGs. Kosari et al. [15–18] investigated domination set, equitable domination set, and new concepts of domination in vague graphs and vague incidence graphs. Pal and Rashmanlou [19, 20] have given several concepts in FGs. Akram et al. [21, 22] introduced certain fuzzy graph structures. Sunitha and Vijayakumar [23] presented some properties of complement on FGs. Muhiuddin et al. [24–26] investigated new results in cubic graphs.
A VGS is concerned with the generalized structure of an FG that expresses more exactness, adaptability, and compatibility to a system when synchronized with systems operating on FGs. Correspondingly, a VGS is capable of focusing on determining the uncertainly combined with the inconsistent and indeterminate information of any real-world problem, in which FGs may not lead to adequate results. Hence, in this paper, we define the certain notions including VGS, SVGS, and vague -cycle and describe these notions by several examples. We study -complement, SC, and SSC in VGSs and investigate some of their properties. Finally, an application of VGS has been presented.
2. Preliminaries
A GS , consists of a nonempty set together with relations on , which are mutually disjoint so that each is irreflexive and symmetric. If for some , , we name it an -edge and write it as “.”
A GS is CGS, if,(i)every edge , appears at least once in (ii)among every pair of nodes in , is an -edge for some ,
Definition 1. (see [1]). A fuzzy subset on a set is a map . A fuzzy binary relation on is a fuzzy subset on . By a fuzzy relation, we mean a fuzzy binary relation given by .
Definition 2. (see [11]). Let be a GS and let be the fuzzy subset of , respectively, so that , , . Then, is an FGS of .
Definition 3. (see [11]). Let be an FGS of a GS . Then, is a PFSSGS of if , for .
Definition 4. (see [11]) The strength of a -path of a FGS is , for .
Definition 5. (see [11]) In a FGS , , , , for any . Also, .
Definition 6. Reference [11] is a -cycle if and only if is a -cycle.
Definition 7. Reference [11] is a fuzzy -cycle if and only if is a -cycle and no unique in so that
Definition 8. (see [3]) A VS is a pair on set X where and are taken as real valued functions which can be defined on so that , .
Definition 9. (see [14]) A VG is a pair , where is a VS on and is a VS on so that and , for .
All the basic notations are shown in Table 1.
3. New Concepts in Vague Graph Structure
Definition 10. is said to be a VGS of a GS , if is a VS on and for every ; is a VS on so that. Note that , and , , , where and are named UVS and underlying -edge set of , respectively.
Example 1. Let be a GS so that , , and . Let , , and be vague subsets of , , and , respectively, so thatThen, is a VGS of drawn in Figure 1.
Definition 11. (i)AVGS is said to be a VSGS of a VGS with UVS , if and , , that is, and for ,(ii) is named a VSSGS of a VGS , if .(iii) is named a VPSSGS of a VGS , if it excludes some edges of .
Example 2. Consider a VGS as shown in Figure 1. LetIt is easy to see that , , and are the VSGS, VSSGS, and VPSSGS of , respectively.
Their corresponding graphs are drawn in Figures 2–4.
Definition 12. Let be a VGS with UVS . Then, there exists a -edge among two nodes and of so that we have(i), , or(ii), , or(iii), , for some .
Definition 13. For a VGS , the support of is described as
Definition 14. -path of a VGS with UVS is a sequence of distinct nodes (except the choice ) so that is a -edge, .
Definition 15. In a VGS with UVS , two nodes and called -connected, if they are connected by a -path, for some .
Definition 16. A VGS with UVS is called to be -strong, if -edges ,for some .
Example 3. Consider the VGS , drawn in Figure 1. Then,(i) are -edges and are -edges(ii) and are and paths, respectively(iii) and are -connected nodes of (iv) is a -strong since and we have
Definition 17. A VGS is named to be strong, if it is -strong, .
Definition 18. A VGS with UVS , is named complete or -complete if(i) is a SVGS(ii), (iii)For each nodes , should be a -edge
Example 4. Suppose drawn in Figure 5, be VGS of the GS where , , and . Then, is a SVGS, since it is both -strong and -strong. Moreover, , , and each nodes in , is either a -edge or a -edge, so is a complete or -complete-VGS as well.
Definition 19. In a VGS with UVS , and -strength of a -path “” are denoted by and , respectively, so that and .
Then, we write strength of the path .
Example 5. In shown in Figure 5, is a -path, and is a -path, we haveThus, strength of -path is , and strength of -path is .
Definition 20. In a VGS with UVS ,(i)-strength of connectedness between and is defined by , where , for and ;(ii)-strength of connectedness among and is described by , where , for and .
Example 6. Let be a VGS of GS as shown in Figure 6 so that , , and . Since , , and , therefore,Thus, we have,Since , , and , therefore,andThus, we haveBy similarity way, we can calculate , , and .
Definition 21. A VGS of a GS is a -cycle, if is an -cycle.
Definition 22. A VGS of a GS is a vague -cycle, for some , if we have(i) is a -cycle;(ii)There is no unique -edge in so that or .
Example 7. VGS in Figure 7 is a -cycle as well as vague -cycle, since is an -cycle and there are two -edges with minimum degree of membership and two -edges with maximum degree of membership of all -edges.
Definition 23. A VGS of GS is isomorphic to a VGS of if a bijective and a permutation on the set so that, , for all , .
Definition 24. A VGS of GS is identical to a VGS of if a bijection so thatand
Example 8. Let and be two VGSs of GS and , respectively, drawn in Figure 8. Here, is isomorphic (not identical) to under the mapping , defined by , , and and a permutation given by , so that
Example 9. Let and be two VGSs of GSs and , respectively, as shown in Figure 9. Here, is identical with under the mapping , defined by , , , , , and so that
Definition 25. Let be a VGS of a GS . Let denote a permutation on the set and the corresponding permutation on , i.e., if , . If for some and , , , then, while is chosen so that and , . Then, VGS denoted by , is named the -complement of VGS.
Example 10. Consider VGS shown in Figure 10 and let be a permutation on the set so that and . Now, for ,Clearly, and , so .So, . Also, for , we haveSo, .
Theorem 1. A -complement of a VGS is a SVGS. In addition, if , for , then all -edges in VGS become -edges in .
Proof. The proof of the first part is clear according to the definition of the -complement of VGS , since for any -edge , and , respectively, have the maximum values ofThat is,for all edges in ; hence, is a SVGS. Now assume on contrary that , but is -edge in with that shows .
By comparing equations (24) and (25), we havethat it is not true because , for some . Therefore, our assumption is wrong and should be a -edge. Thus, we get if , then, all -edges in VGS become -edges in , for .
Definition 26. Let be a VGS and be a permutation on the set . Then,(i) is SC, if ,(ii) is SSC, if it is identical to .(iii) is TSC if , permutations on the set .(iv) is TSSC, if it is identical to , permutations on the set .
Theorem 2. A VGS is strong if is TSC.
Proof. Assume be a SVGS and be a permutation on the set . If , then, by Theorem 1, all -edges in become -edges in . Also, is strong, so,Then, is isomorphic to , under the identity mapping and a permutation , so thatandfor all . This holds for all the permutation on the set . Thus, is TSC. Conversely, let be any permutation on the set and and be isomorphic. According to the definitions of -complement and isomorphism, we getfor all , . Hence, is a SVGS.
Theorem 3. If GS is TSSC and is a VS of with constant fuzzy mapping and , then, a SVGS of is TSSC.
Proof. Consider a SVGS of a GS . Assume that is TSSC and that for some constants , is a VS of so that , , . Then, we have to prove that is TSSC. Let be an arbitrary permutation on the set and . Since is TSSC, so a bijection so that for each -edge in , is an -edge in . Consequently, for every -edge in , is a -edge in . From the definition of and the definition of SVGS ,andfor all , , which shows is SSC. Hence, is TSSC, since is arbitrary.
Example 11. A VGS of GS as shown in Figure 11 is TSSC.
4. Application
Providing services to the people in the shortest possible time is one of the most important issues that governments strive to do in the best possible way. Unfortunately, in the past, due to the lack of medical and welfare services as well as the lack of educational centers, people had to travel to distant cities to provide these necessary facilities and equipment, and in this way, unfortunately, they incurred a lot of costs. But today, with the help of governments as well as the equipping of medical and welfare centers, people can move to their nearest neighboring city to benefit from these facilities. A vague graph structure helps us to understand which route is more appropriate for a particular service. It also shows us which service is increasing so that governments can strengthen it in the same way and also try to increase the rest of the services so that people can benefit from them. Most importantly, a VGS can also be useful to the mayor of the area, as it helps the mayor make appropriate policies for the city to develop appropriate services. For example, if one of the welfare services in that area is more than other services (the number of people referring to that welfare center is higher), then, the VGS shows this issues and as a result, the mayor takes the necessary measures to maintain and increase this opportunity. Suppose is a set consisting of 7 cities as , and is a VS on as Table 2.
In Table 2, true membership value indicates the amount of welfare, educational, medical, and peace facilities in that area, and the false-membership value reveals the lack of necessary facilities in each of the medical services. In Tables 3–9, we show the membership rate of various welfare services on the route between each pair of cities. The relations that considered on set are as follows:
, , , , and so that is a graph structure. Each object in a particular relation indicates the amount of services provided between the two cities. In these relations, each element is determined by its corresponding membership rate, and the corresponding sets are FSs on , , , and , respectively, which are shown by , , , , and , respectively.
Suppose that
Now, corresponding vague sets are
It is obvious that is a VGS and is denoted in Figure 12. In the VGS shown in Figure 12, each edge represents the most services that occur on the connecting route between the two cities. For example, most services on the way from Babolsar to Bahnamir are grocery services, and their membership value are . It is clear that the city of Sari has the highest membership in terms of banking services, which means that this city is the most sensitive city for providing banking services. According to the VGS in Figure 12, the most common services is car dealership. Therefore, we conclude that government should identify places that are suitable for providing services between cities so that people can benefit from these services at the lowest cost and in the fastest possible time.
5. Conclusion
Fuzzy graph has various uses in modern science and technology, especially in the fields of neural networks, computer science, operation research, and decision making. VGS have more precision, flexibility, and compatibility, as compared to the fuzzy graphs. Today, VGSs play an important role in social networks and allow users to find the most effective person in a group or organization. So, in this paper, we presented certain notions, including VGS, SVGS, and vague -cycle and illustrated these notions by several examples. We investigated -complement, SC, SSC, and TSSC in VGS and studied some of their properties. Finally, an application of VGS has been introduced. In our future work, we will introduce neighborly irregular, highly irregular, strongly irregular, and edge irregularity in VGSs and investigate some of their properties.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.