Abstract
We apply the concept of -polar fuzzy sets to graph structures. We introduce certain concepts in -polar fuzzy graph structures, including strong -polar fuzzy graph structure, -polar fuzzy -cycle, -polar fuzzy -tree, -polar fuzzy -cut vertex, and -polar fuzzy -bridge, and we illustrate these concepts by several examples. We present the notions of -complement of an -polar fuzzy graph structure and self-complementary, strong self-complementary, totally strong self-complementary -polar fuzzy graph structures, and we investigate some of their properties.
1. Introduction
Graph theory has applications in many areas of computer science, including data mining, image segmentation, clustering, image capturing, and networking. A graph structure, introduced by Sampathkumar [1], is a generalization of undirected graph which is quite useful in studying some structures including graphs, signed graphs, and graphs in which every edge is labeled or colored. A graph structure helps to study the various relations and the corresponding edges simultaneously.
A fuzzy set [2] is an important mathematical structure to represent a collection of objects whose boundary is vague. Fuzzy models are becoming useful because of their aim in reducing the differences between the traditional models used in engineering and science. Nowadays fuzzy sets are playing a substantial role in chemistry, economics, computer science, engineering, medicine, and decision-making problems. In 1997, Zhang [3] generalized the idea of a fuzzy set and gave the concept of bipolar fuzzy set on a given set as a map which associates each element of to a real number in the interval In , Chen et al. [4] introduced the idea of -polar fuzzy sets as an extension of bipolar fuzzy sets and showed that bipolar fuzzy sets and -polar fuzzy sets are cryptomorphic mathematical notions and that we can obtain concisely one from the corresponding one in [4]. The idea behind this is that “multipolar information” (not just bipolar information which corresponds to two-valued logic) exists because data for a real world problem are sometimes from agents . For example, the exact degree of telecommunication safety of mankind is a point in because different person has been monitored different times. There are many examples such as truth degrees of a logic formula which are based on logic implication operators , similarity degrees of two logic formula which are based on logic implication operators , ordering results of a magazine, ordering results of a university, and inclusion degrees (accuracy measures, rough measures, approximation qualities, fuzziness measures, and decision preformation evaluations) of a rough set.
Kauffman [5] gave the definition of a fuzzy graph in on the basis of Zadeh’s fuzzy relations [6]. Rosenfeld [7] discussed the idea of fuzzy graph in 1975. Further remarks on fuzzy graphs were given by Bhattacharya [8]. Several concepts on fuzzy graphs were introduced by Mordeson and Nair [9]. In 2011, Akram introduced the notion of bipolar fuzzy graphs in [10]. Alshehri and Akram [11] introduced the concept of bipolar fuzzy competition graphs. In 2015, Akram and Younas studied certain types of irregular -polar fuzzy graphs in [12]. Akram and Adeel studied -polar fuzzy line graphs in [13]. Akram and Waseem introduced certain metrics in -polar fuzzy graphs in [14]. Dinesh [15] introduced the notion of a fuzzy graph structure and discussed some related properties. Akram and Akmal [16] introduced the concept of bipolar fuzzy graph structures. In this paper, we apply the concept of -polar fuzzy sets to graph structures. We introduce certain concepts in -polar fuzzy graph structures, including strong -polar fuzzy graph structure, -polar fuzzy -cycle, -polar fuzzy -tree, -polar fuzzy -cut vertex, and -polar fuzzy -bridge, and we illustrate these concepts by several examples. We present the notions of -complement of an -polar fuzzy graph structure and self-complementary, strong self-complementary, totally strong self-complementary -polar fuzzy graph structures, and we investigate some of their properties.
2. Preliminaries
In this section, we review some basic concepts that are necessary for fully benefit of this paper.
In , Zadeh [2] introduced the notion of a fuzzy set as follows.
Definition 1 (see [2, 6]). A fuzzy set in a universe is a mapping . A fuzzy relation on is a fuzzy set in . Let be a fuzzy set in and fuzzy relation on . We call a fuzzy relation on if .
Definition 2 (see [16]). is called a bipolar fuzzy graph structure (BFGS) of a graph structure (GS) if is a bipolar fuzzy set on and for each is a bipolar fuzzy set on such that Note that for all and , , where and are called underlying vertex set and underlying -edge sets of , respectively.
Definition 3 (see [16]). Let be a BFGS of a GS Let be any permutation on the set and the corresponding permutation on , if and only if
If for some and then , while is chosen such that
And BFGS , denoted by , is called the -complement of BFGS
Definition 4 (see [4]). An -polar fuzzy set (or a -set) on is exactly a mapping
Note that a -set is an -set. An -set on the set is a synonym of a mapping , where is a lattice. So, is considered to be a partial order set with the point-wise order , where is an arbitrary ordinal number, is defined by for each , and is the th projection mapping (). When , an -set on will be called a fuzzy set on .
Definition 5 (see [14]). Let be an -polar fuzzy subset of a nonempty set . An -polar fuzzy relation on is an -polar fuzzy subset of defined by the mapping such that, for all , , where denotes the th degree of membership of the vertex and denotes the th degree of membership of the edge .
Definition 6 (see [4, 14]). An -polar fuzzy graph is a pair , where is an -polar fuzzy set in and is an -polar fuzzy relation on such that for all .
We note that for all for all . is called the -polar fuzzy vertex set of and is called the -polar fuzzy edge set of , respectively. An -polar fuzzy relation on is called symmetric if for all .
3. Certain Concepts in -Polar Fuzzy Graph Structures
Definition 7. Let be a graph structure (GS). Let be an -polar fuzzy set on and an -polar fuzzy set on such thatfor all , , and for . Then is called an -polar fuzzy graph structure (-PFGS) on where is the -polar fuzzy vertex set of and is the -polar fuzzy i-edge set of .
Definition 8. Let and be two -PFGSs of a GS . Then is called an -polar fuzzy subgraph structure of , if is called an -polar fuzzy induced subgraph structure of by a set , if is called an -polar fuzzy spanning subgraph structure of , if and
Example 9. Consider a graph structure such that , , and Let , , and be 4-polar fuzzy sets on , , and , respectively, defined by Tables 1 and 2.
By simple calculations, it is easy to check that is a 4-polar fuzzy graph structure of as shown in Figure 1. Note that we represent as in all tables and the figures.
A -polar fuzzy induced subgraph structure of by and a -polar fuzzy spanning subgraph structure of (Figure 1) are shown in Figure 2.
Definition 10. Let be an -PFGS. Then is called an -polar fuzzy -edge or simply a -edge, if for at least one and we write that Consequently, support of is defined byAn -polar fuzzy -path, in -PFGS , is a sequence of vertices (distinct except the choice ) in such that each is an -polar fuzzy -edge for
Definition 11. A -strong -PFGS is an -polar fuzzy graph structure that satisfiesfor all is a strong -polar fuzzy graph structure, if it is -strong for all .
Definition 12. An -PFGS is called complete, if the following three conditions hold:(i)polar fuzzy graph structure is strong.(ii).(iii)For each pair of vertices , is a -edge for some .
Example 13. Consider a graph structure such that , , and Let , , and be 4-polar fuzzy sets on , and , respectively, defined by Tables 3 and 4.
By simple calculations, it is easy to check that is a strong and complete 4-polar fuzzy graph structure of , as shown in Figure 3. is a 4-polar fuzzy -path consisting of all -edges constituting .
Definition 14. Let be an -PFGS and let . The j-strength or j-gain of a -path , denoted by , is defined by
Example 15. Consider the -PFGS , shown in Figure 3, in which is a 4-polar fuzzy -path. Then -gain, for , of this path is given by , , , and .
Definition 16. Let be an -polar fuzzy graph structure and let . Then -strength of connectedness between two vertices and is defined by , where for and for and
Example 17. Consider the -PFGS as shown in Figure 3. All the calculated -strengths of connectedness between any two vertices of , for , are given in Table 5.
Definition 18. An -polar fuzzy -cycle is an -PFGS such that(i)() is an -cycle (it is called a -cycle),(ii)There is no unique -edge in such that for any
Example 19. Let be the -PFGS, shown in Figure 4; then is a -polar fuzzy -cycle.
Definition 20. Let be an -polar fuzzy graph structure and let be a vertex in . Let be an -polar fuzzy subgraph structure of induced by , such that for Then is called an -polar fuzzy -cut vertex for some , if for some and is called an -polar fuzzy -cut vertex for some , if it is -cut vertex for all
Definition 21. Let be an -polar fuzzy graph structure and an edge in Let be an -polar fuzzy spanning subgraph structure of such thatThen is an -polar fuzzy -bridge for some , if for some And is called an -polar fuzzy -bridge for some , if it is -bridge for all
Example 22. Consider the -PFGS as shown in Figure 3. The vertex is a -polar fuzzy -cut vertex since and , where is the -polar fuzzy subgraph structure of induced by .
The edge is a -polar fuzzy -bridge since , , , and , where is the -polar fuzzy spanning subgraph structure of that excludes the -edge .
Definition 23. An -PFGS is a -tree if () is an -tree.
Definition 24. An -PFGS is an -polar fuzzy -tree if there is an -polar fuzzy spanning subgraph structure such that is a -tree and for every edge not belonging to If the above condition holds for , is called a -tree.
Example 25. Consider the -PFGS as shown in Figure 5.
Then is a -polar fuzzy -tree since , where is the -polar fuzzy spanning subgraph structure of that excludes the -edge . Also is a -tree.
Definition 26. An -PFGS of a graph structure is said to be isomorphic to an -PFGS of a graph structure , if there exists a bijection and a permutation on the set such that for
Example 27. Two isomorphic 4-PFGSs and are shown in Figure 6. This isomorphism holds under the permutation and the mapping , defined by , and
Definition 28. An -PFGS is said to be identical to an -PFGS if there exists a bijection , such that for
Example 29. Two 4-PFGSs and , identical under the mapping , defined by , and , are shown in Figure 7.
Definition 30. Let be an -polar fuzzy graph structure. Let be any permutation on the set and If for some and for then , where is chosen such that, for each , Then an -PFGS constructed with and -polar fuzzy relations () on , denoted by , is called the -complement of -polar fuzzy graph structure .
Proposition 31. A -complement of an -polar fuzzy graph structure is always a strong -PFGS. Moreover, if for , then all -edges in -PFGS become -edges in
Proof. From the definition of -complement of an -PFGS , We can see that , and SoMoreover, the maximum value of occurs when its negative part () becomes zero, and it is zero when and is a -edge in . Therefore, for
Hence is a strong -PFGS and every -edge in becomes -edge for
Remark 32. If is an -polar fuzzy graph structure of a GS then -complement of is a strong -polar fuzzy graph structure of -complement of .
Definition 33. Let be an -polar fuzzy graph structure and be any permutation on the set . Then(i) is called self-complementary if is isomorphic to ,(ii) is called strong self-complementary if is identical to .
Definition 34. Let be an -polar fuzzy graph structure. Then(i) is called totally self-complementary if is isomorphic to for every permutation on the set ,(ii) is called totally strong self-complementary if is identical to for every permutation on the set
Theorem 35. An -PFGS is strong if and only if it is totally self-complementary.
Proof. We suppose that is a strong -PFGS of a GS and is any permutation on the set
By Proposition 31, is strong and all -edges in become -edges for So Using the identity mapping , we get , and That is,Hence is isomorphic to under the identity mapping and a permutation . Since was an arbitrary permutation on , is totally self-complementary.
Conversely suppose that is a totally self-complementary -PFGS and we have to prove that is strong -PFGS.
Since and are isomorphic for all permutations on , therefore by definition of isomorphism , for all Hence is a strong -PFGS.
This completes the proof.
Theorem 36. Let be a strong -PFGS on a GS . If is totally strong self-complementary and is an -polar fuzzy set on such that for each assigns a constant value to every , then is totally strong self-complementary.
Proof. We assume that is totally strong self-complementary and for all , where is a constant for each
Since is totally strong self-complementary, for any permutation with , there exists a bijection , such that, for every -edge in , (an -edge in ) is an -edge in . Consequently, for every -edge in , (a -edge in ) is a -edge in . Moreover, and is a strong -PFGS, so we get This shows that is identical to . Hence is totally strong self-complementary, since was an arbitrary permutation. This completes the proof.
Converse of Theorem 36 does not hold because every strong and totally strong self-complementary -PFGS does not necessarily include an -polar fuzzy vertex set such that for each assigns a constant value to every This can be observed in -PFGS shown in Figure 9.
Example 37. There are no other totally self-complementary -PFGSs than strong -polar fuzzy graph structures. So all strong -PFGSs are the examples of totally self-complementary -PFGSs.
Example 38. An -PFGS, shown in Figure 8, is strong self-complementary but not totally strong self-complementary because this -PFGS is identical to its -complement only when .
An -PFGS, shown in Figure 9, is totally strong self-complementary.
4. Conclusions
A graph structure is a useful tool in solving the combinatorial problems in different areas of computer science and computational intelligence systems. It helps to study various relations and the corresponding edges simultaneously. Sometimes information in a network model is based on multiagent, multiattribute, multiobject, multipolar information or uncertainty rather than a single bit. An polar fuzzy model is useful for such network models which gives more and more precision, flexibility, and comparability to the system as compared to the classical, fuzzy, and bipolar fuzzy models. We have introduced certain concepts in -polar fuzzy graph structures. We are extending our work to (1) neutrosophic graph structures, (2) intuitionistic neutrosophic soft graph structures, (3) roughness in graph structures, and (4) intuitionistic fuzzy soft graph structures.
Competing Interests
The authors declare that there is no conflict of interests regarding the publication of this article.