Abstract
Molodtsov’s theory of soft sets is free from the parameterizations insufficiency of fuzzy set theory. Type-2 soft set as an extension of a soft set has an essential mathematical structure to deal with parametrizations and their primary relationship. Fuzzy type-2 soft models play a key role to study the partial membership and uncertainty of objects along with underlying and primary set of parameters. In this research article, we introduce the concept of fuzzy type-2 soft set by integrating fuzzy set theory and type-2 soft set theory. We also introduce the notions of fuzzy type-2 soft graphs, regular fuzzy type-2 soft graphs, irregular fuzzy type-2 soft graphs, fuzzy type-2 soft trees, and fuzzy type-2 soft cycles. We construct some operations such as union, intersection, AND, and OR on fuzzy type-2 soft graphs and discuss these concepts with numerical examples. The fuzzy type-2 soft graph is an efficient model for dealing with uncertainty occurring in vertex-neighbors structure and is applicable in computational analysis, applied intelligence, and decision-making problems. We study the importance of fuzzy type-2 soft graphs in chemical digestion and national engineering services.
1. Introduction
Fuzzy set theory has its remarkable origin to the work of Zadeh [1] in 1965 to interact with vagueness and imprecision between absolute true and absolute false. The range of the values in a fuzzy set lies in [0, 1]. This remarkable discovery of fuzzy set theory paved a different way for dealing with uncertainties in various domains of science and technology.
Graph theory is moving quickly into the mainstream of mathematics, primarily due to its applications in engineering, communication networks, computer science, and artificial intelligence. In 1973, Kauffmann [2] introduced the notion of fuzzy graph, which is based on Zadeh’s fuzzy relation [3]. Another elaborated definition of fuzzy graph was introduced by Rosenfeld [4]. Bhattacharya [5] subsequently gave some helpful results on fuzzy graphs and some operations on fuzzy graph theory were explored by Mordeson and Nair [6]. Many researchers studied fuzzy graphs in recent decades [7–9].
However, the theory of fuzzy sets has inadequacy to deal with parametrization tool. Soft set theory proposed by Molodtsov [10] has the ability to cope with this difficulty and is defined as a pair , where is a mapping given by . Soft sets have been generalized to numerous directions beginning with Maji et al. [11, 12] who introduced fuzzy soft sets and Ahmad and Kharal [13] discussed some properties of fuzzy soft sets. In algebraic structures, soft sets and their hybrid models based on fuzzy soft sets, generalized fuzzy soft sets, rough soft sets, and soft rough sets have been implemented effectively [14–19]. Sarwar [20] elaborated the notion of rough graph and discussed decision-making approaches based on rough numbers and rough graphs. Akram and Nawaz [21] introduced the concepts of fuzzy soft graphs (named as fuzzy type-1 soft graph), vertex-induced soft graphs, and edge-induced soft graphs and also discussed some operations on soft graphs. Akram and Zafar [22] introduced various hybrid models based on fuzzy sets, soft sets, and rough sets. Further, Akram in cooperation with other researchers [23–26] discussed various applications and extensions of graph theory to study different types of uncertainties in real-world problems. Nowadays, researchers are actively working on interval type-2 fuzzy arc lengths [27], trapezoidal interval type-2 fuzzy soft sets [28], total uniformity of graph under fuzzy soft information [29], fuzzy soft cycles [30], and fuzzy soft coverings.
All these existing models have the same restriction that one cannot freely select the parameters. That is, if a correspondence or association occurs between parameters, then none of these models can solve the problems completely. Chatterjee et al. [31] proposed the concept of type-2 soft sets to deal with the correspondence between parameters, which is a generalization of Molodtsov’s soft sets (called type-1 soft sets). Type-2 soft sets reparameterize the already parameterized crisp sets and thus have more freedom and effectiveness in dealing with imprecision as compared to type-1 soft sets. Hayat et al. [32–34] introduced vertex-neighbors-based type-2 soft sets, type-2 soft graphs, and irregular type-2 soft graphs and presented certain types of type-2 soft graphs.
The motives of this study are as follows: (1)Soft sets and their hybrid models are used to deal with uncertainty based on parametrization tool. The correspondence, association, or relation occurring among parameters cannot be discussed with existing approaches. Type-2 soft models tackle this difficulty and present a mathematical approach to reparameterize the existing soft models. To deal with partial membership of objects, the main focus of this study is to introduce a hybrid model by combining fuzzy set theory with type-2 soft sets.(2)Graph theory is an essential approach to study relations among objects using a figure consisting of vertices and lines joining these vertices. But there is an information loss in graphical models whether the objects are fully related or partially related, that is, uncertain and parameterized relations among objects. To handle this information loss, there is a need to represent the graphical models under fuzzy type-2 soft environment.
The main contribution of this study is as follows:(1)The present study introduces the mathematical approaches of vertex-neighbors-based type-2 soft set and vertex-neighbors-based type-2 soft graphs under fuzzy environment. The notions of fuzzy type-2 soft graphs, regular fuzzy type-2 soft graphs, irregular fuzzy type-2 soft graphs, fuzzy type-2 soft trees, and fuzzy type-2 soft cycles are discussed with certain operations and numerical examples.(2)The importance of presented concepts is studied with an application in chemical digestion and national engineering services.
2. Preliminaries
The term crisp graph on a nonvoid universe (named as set of vertices) is defined as a pair , where is named as set of edges. Crisp graph is a special case of the fuzzy graph with each vertex and edge of having degree of membership 1. A soft graph corresponding to a crisp graph is a parameterized family of subgraphs of . A soft graph on a nonempty set is a tuple such that, for each , is a graph, where and .
Definition 1 (see [31]). Let be a soft universe and let be the set of all T1SSs over . Then a mapping is called a type-2 soft set (T2SS) over and it is denoted by . For all is a T1SS such that , where and . We refer to the parameter set as the “primary set of parameters” although the collection of parameters denoted by is called “underlying set of parameters.”
Definition 2 (see [32]). Suppose that is a simple graph. Suppose that and is the set of all T1SSs over . Suppose that is a T2SS over . Then a mapping is said to be a T2SS over and is denoted as . For every vertex , is a T1SS, where and can be explained as , This T2SS is said to be a vertex-neighbors type-2 soft set (VN-T2SS) over .
Definition 3 (see [32]). Suppose that is a simple graph. Suppose that and is the set of all T1SSs over . Suppose that is a VN-T2SS over . Then a mapping is said to be a T2SS over and is denoted as . For every vertex , is a T1SS, where and can be explained as . This T2SS is said to be a VN-T2SS over .
We present the notations that are used in this research article in Table 1.
3. Fuzzy Type-2 Soft Graphs
We refer to Maji’s [11] fuzzy soft set as fuzzy type-1 soft set (FT1SS). Consider as a set of parameters that have a random nature (characterization of object, some functions, numeric values, etc.). Consider as a universal set and the class of all FT1SSs over will be indicated by . Recently, researchers have shown attraction to the application of fuzzy soft sets in science, advance technology, and decision problems. Fuzzy type-2 soft sets are considered as a generalized form of fuzzy soft set. Consider as a universal set and as the set of parameters. Fuzzy type-2 soft set is defined as follows.
Definition 4. Let be a fuzzy soft universe and let be the collection of all FT1SSs over . Then a mapping , , is called a fuzzy type-2 soft set (FT2SS) over and it is denoted by . In this case, corresponding to each parameter , is FT1SS. Thus, for each , there exists a FT1SS such that , where and . In this case, we refer to the parameter set as the “primary set of parameters,” while the set of parameters is known as the “underlying set of parameters.”
Definition 5. Let be a fuzzy graph. The set of neighbors of an element is denoted by and defined by . Then .
Definition 6. Let be a fuzzy graph. Suppose that and is the set of all FT1SSs over . Suppose that is a FT2SS over . Then a mapping is said to be a FT2SS over and is denoted as . For every vertex , is a FT1SS and can be explained as . This FT2SS is said to be a vertex-neighbors fuzzy type-2 soft set (VN-FT2SS) over .
Definition 7. Let be a fuzzy graph. Suppose that and is the set of all T1SSs over . Suppose that is a FT2SS over . Then a mapping is said to be a FT2SS over and is denoted as . For every vertex , is a FT1SS and can be explained as . This FT2SS is said to be a VN-FT2SS over .
and are FT1SS over and , respectively. If represent a fuzzy graph in fuzzy type-2 soft graph , then is called FT1SG.
Definition 8. A 5-tuple is called a fuzzy type-2 soft graph (FT2SG) if it satisfies the following conditions:(a) is a fuzzy graph.(b) is a nonempty set of parameters.(c) is a VN-FT2SS over .(d) is a VN-FT2SS over .(e)FT1SS corresponding to represents a VN-fuzzy type-1 soft graph (FT1SG).A FT2SG can also be defined by , where such that = .
Example 1. Let be a fuzzy graph as shown in Figure 1. Let . Suppose that and are two FT2SSs over and , respectively. We haveDefine and . Then FT2SSs and are defined as follows:Fuzzy type-2 soft graph is shown in Figure 2.
Definition 9. Let be a fuzzy type-2 soft graph; the complement of is denoted by and defined by , where is the complement of FT1SG corresponding to for all .
Example 2. Let be a fuzzy graph as shown in Figure 3.
Let , and .
Let and be two FT2SSs over and , respectively. We haveDefineThe FT2SSs and are defined as follows:Then and are defined as follows:The complement of is a FT2SG such that is the complement of FT1SG corresponding to and is the complement of FT1SG corresponding to as shown in Figure 4.
Definition 10. Let be a FT2SG; is said to be a regular FT2SG if FT1SG corresponding to is a regular FT1SG .
Proposition 1. If is a regular FT2SG, then is a regular FT2SG.
Proof. Let be a regular FT2SG. Suppose that is a FT1SG corresponding to ; then must be a regular fuzzy graph. As we know that complement of a regular graph is regular, is also a regular fuzzy graph. It provides FT1SG corresponding to a for all being regular FT1SG. Thus, is a regular FT2SG of .
Definition 11. Let be a FT2SG; is said to be an irregular FT2SG if FT1SG corresponding to is an irregular FT1SG .
Example 3. Let be a fuzzy graph as shown in Figure 5. Let , and .
Let and be two FT2SSs over and , respectively. We haveDefineThe FT2SSs and are defined as follows:Then is an irregular FT2SG as shown in Figure 6.
Proposition 2. If is a regular fuzzy graph, then every FT2SG of is not necessarily a regular FT2SG.
Definition 12. Let be a FT2SG; is called a neighborly irregular FT2SG if FT1SGs corresponding to are neighborly irregular FT1SG .
Example 4. Let be a fuzzy graph as shown in Figure 7. Let , and .
Let and be two FT2SSs over and , respectively. We haveDefineThe FT2SSs and are defined as follows:Then is a neighborly irregular FT2SG as shown in Figure 8.
Definition 13. Let be a FT2SG and is a FT1SG for all . An edge in is said to be a FT2S bridge if its removal disconnects for all .
Definition 14. Let be a FT2SG and is a FT1SG for all . A vertex in is said to be a FT2S cut-vertex if its removal disconnects for all .
Definition 15. Let be a FT2SG; is called a fuzzy type-2 soft tree (FT2ST) if FT1SGs corresponding to are FT1STs for all .
Example 5. Let be a fuzzy graph as shown in Figure 9. Let , and .
Let and be two FT2SSs over and , respectively. We haveDefineThe FT2SSs and are defined as follows:Then is a FT2ST as shown in Figure 10. It can also be defined as VN-type-2 soft tree.
Theorem 1. Let be a FT2SG and is a FT1SG for all . If is a FT1SG with vertices, then will not be a complete FT2SG.
Proof. Let be a FT2SG and is a FT1SG for all . On the contrary, assume that is a complete FT2SG; then each will also be complete. Let be arbitrary nodes of joined by a line . Since having vertices of is a FT1SG, then a minimum one vertex which is connected to by an edge and to by an edge as be a complete fuzzy graph. Then there is a cycle . Therefore, cannot be a FT1ST, which is opposite to the fact that is a connected FT1SG of FT2SG. So, is not a complete FT2SG.
Definition 16. Let be a FT2SG and is a FT1SG for all . Then is called a fuzzy type-2 soft forest if consists of several disjoined fuzzy trees .
Definition 17. Let be a FT2SG; is said to be a FT2SC if FT1SG corresponding to is a fuzzy type-1 soft cycle, for all .
Example 6. Let be a fuzzy graph as shown in Figure 11, whereLet , and .
Let and be two FT2SSs over and , respectively. We haveDefineThe FT2SSs and are defined as follows:We can check that is a FT2SC as shown in Figure 12. It is also defined as a fuzzy VN-type-2 soft cycle.
Example 7. Let be a fuzzy graph as shown in Figure 13. Let , , and . Let and be two FT2SSs over and , respectively. We haveDefine .
The FT2SSs and are defined as follows: and are FT1SGs as shown in Figure 14. We can see that and are all not trees. Hence is not a FT2ST and is also not a FT2SC.
Proposition 3. Every fuzzy type-2 soft cycle is a regular fuzzy type-2 soft cycle.
Proof. Let be a FT2SC. Let be a T1FSC corresponding to for every . Then, is a cycle . We know that cycle is a path that is closed and every vertex of cycle is of degree 2; this signifies that is a regular fuzzy graph for all . Therefore, is a regular FT1SG, for all . Hence is a regular FT2SG.
Definition 18. Let and be two FT2STs. is a fuzzy type-2 soft subtree (FT2SST) of if(i)(ii)For each , FT1ST corresponding to is a fuzzy type-1 soft subtree (FT1SST) of FT1ST corresponding to
Example 8. Let be a fuzzy graph as shown in Figure 15, whereLet . Let and be two FT2SSs over and , respectively. We haveDefine and .
Then FT2SSs and are defined as follows: is a FT2ST as shown in Figure 16.
Let and be two FT2SSs over and , respectively. We haveDefine and .
FT2SSs and are defined as follows: is a FT2SST of as shown in Figure 17. We can see that and . Hence, is a FT2SST of .
Theorem 2. Let and be two FT2STs. Then is said to be a FT2SST of if and only if and .
Proof. Let be a FT2SST of . Then, by using the definition of FT2SST,(i)(ii)For all , FT1ST corresponding to is a FT1SST of FT1ST corresponding to Since FT1ST corresponding to is a FT1SST of FT1ST corresponding to for all , we have and . Conversely, we have and . As is a fuzzy type-2 soft tree, fuzzy type-1 soft set corresponding to forms a FT1ST of for all . Also, is a fuzzy type-2 soft tree, and fuzzy type-1 soft set corresponding to forms a FT1ST of for all . This implies that FT1ST corresponding to is a FT1SST of FT1ST corresponding to for all . Hence, is a FT2SST of .
Definition 19. Let and be two FT2STs. The union of and is denoted by , where , such thatwhere for all relates to the fuzzy type-1 soft union between the relevant FT1STs corresponding to and , respectively. It can be written as .
Theorem 3. Let and be two FT2STs with . Then is a FT2ST.
Proof. Let and be two FT2STs. The union of and is denoted by , where is defined :where for all relates to the fuzzy type-1 soft extended union among the relevant FT1STs corresponding to and , respectively, and for all relates to the fuzzy type-1 soft union between the relevant FT1STs corresponding to and , respectively. Since is a FT2ST, FT1ST corresponding to is a FT2ST for all .
Since is a FT2ST, FT1ST corresponding to is a FT2ST for all . It is given that . Thus, is a FT2ST.
Definition 20. Let and be two FT2STs. The intersection of and is denoted by , where such thatwhere for all relates to the fuzzy type-1 soft intersection between the relevant FT1STs corresponding to and , respectively.
It can be written as .
Example 9. Let be a fuzzy graph as shown in Figure 18. Let . It can be written as . Let and be two FT2SSs over and , respectively. We haveDefine and .
The FT2SSs and are defined as follows:Let and be two FT2SSs over and , respectively. We haveThe FT2SSs and are defined as follows:Then and are FT2STs as shown in Figure 19. By the definition of intersection of FT2STs, and .
Therefore, is a FT2ST as shown in Figure 20.
Definition 21. Let and be two FT2STs. The AND operation of and is denoted by such that . for all is the fuzzy type-1 soft AND operation between the relevant FT1SGs corresponding to and , respectively.
Example 10. Let be the fuzzy graph as shown in Figure 21, whereLet .
Let and be two FT2SSs over and , respectively. We haveThe FT2SSs and are defined as follows:Then is a FT2ST. Let and be two FT2SSs over and , respectively. We haveDefine , , is a FT2ST. The AND operation of and is defined as follows:The AND operation of and is shown in Figure 22.
Definition 22. Let and be two FT2STs. The OR operation of and is denoted by such that . for all is the fuzzy type-1 soft OR operation between the relevant FT1SGs corresponding to and , respectively.
4. Applications of Fuzzy Type-2 Soft Graphs
In this section, we apply the concept of fuzzy type-2 soft graphs to decision-making problems in chemical digestion and national engineering services. The selection of a suitable object problem can be considered as a decision-making problem, in which final identification of object is decided on a given set of information. A detailed description of the algorithm for the selection of most suitable object based on available set of parameters is given in Algorithm 1 and the flow chart shown in Figure 23; purposed algorithm can be used to find out the best correspondence relationship between the neighboring objects in the decision-making problem. This method can be applied in various domains for multicriteria selection of objects.
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4.1. Determination of Dominant Food Components in Chemical Digestion
We present an application of FT2SG in chemical digestion and discuss how to apply FT2SG in chemical digestion of spinach. Spinach is generally composed of carbohydrates, protein, lipids, minerals, vitamins, and nucleic acids. We mainly focused on the digestion of carbohydrates, proteins, lipids, and nucleic acids, which is carried out by a variety of salivary enzymes and the enzymes present in other parts of digestive system; that is, amylase, pepsin, and trypsin are released as a result of involuntary signal generated by our body to digest the food. When 25 g of spinach is taken, it contains carbohydrates , protein (), lipids , nucleic acids , involuntary signal (0.3), pepsin (0.2), amylase (0.2), and trypsin (0.1), represented as vertices donated by , respectively. “Chemical digestion” is the enzyme-mediated, hydrolysis method that converts large macronutrients into smaller molecules.(i)Carbohydrate mostly comprises amylose and glycogen. Long carbohydrates chains are broken down into disaccharides which are decomposed by amylase enzyme.(ii)Proteins are usually broken down into amino acids by peptidase enzyme as well as trypsin and chymotrypsin.(iii)Lipids are hydrolyzed by pancreatic lipase enzyme.(iv)Nucleic acids, that is, DNA and RNA, are hydrolyzed by pancreatic nuclease.(v)Involuntary signal is generated by the brain in order to carry out chemical digestion in the digestive system.
Protein digestion occurs in stomach and duodenum by the action of three primary enzymes.(i)Pepsin, disguised by abdomen(ii)Trypsin, disguised through pancreas(iii)Amylase, disguised through saliva and pancreas
Note that the values of pepsin, trypsin, amylase, and involuntary signal are supposed as we cannot calculate the amounts of these products released as a result of consumption of little amount of food through previous literature findings.
Consider
In fuzzy graph as shown in Figure 24, edges represent the amount of energy utilized by the body in order to carry out the digestion process. Let represent the amounts of carbohydrates and protein released when 25 g of spinach is consumed. We have .
Let and be two FT2SSs over and , respectively. We have
The FT2SSs and are defined as follows:
The fuzzy type-2 soft graph is shown in Figure 25.
The tabular representations of resultant vertex-neighbors fuzzy graphs shown in Figure 26 corresponding to the parameter with the choice values for all are given in Tables 2 and 3.
The decision value is from the choice value of fuzzy type-2 soft graphs for . The prominent food components are as carbohydrates and as lipids as carbohydrates are consumed as sugar and lipids are consumed as fats. Clearly, the dominant food components are or .
4.2. Water Supply for National Engineering Services
We present the application of fuzzy type-2 soft graph in the National Engineering Services Pakistan (NESPAK). The National Engineering Services Pakistan is a Pakistani multinational state-owned corporation that provides construction, management, and consulting services globally. Every government project has something to do with NESPAK at some time of its planning or implementation. In fuzzy graph as shown in Figure 27, vertices represent some important projects.
NESPAK provides engineering services for these projects, the membership value of a vertex showing the working capability of the relevant project and values of edges represents the strength of the relationship between different projects to complete the tasks.
Now, we take two important projects Plumbing and Solid Waste Management named as , respectively, and . The vertex-neighbors of these selected projects are and . Let and be two FT2SSs over and , respectively. We have
The FT2SSs and are defined as follows:
FT1SGs corresponding to and , respectively, are shown in fuzzy type-2 soft graph 28. (Figure 28)
The tabular representations of resultant vertex-neighbors fuzzy graphs and shown in Figure 29 with the choice values for all are given in Tables 4 and 5.
The decision value is , from the choice value of fuzzy type-2 soft graphs for . The optimal project is “ water supply.” So, NESPAK provides the best engineering services to the project of “water supply.”
Advantages of the Proposed Method.
The advantages of the proposed method based on FT2SGs are as follows:(1)The method can be effectively used to handle uncertainty and vagueness with correspondence, assertion, and relations among parameters.(2)The proposed method incorporates parametrization tool with fuzzy information to effectively handle more uncertain conditions and errors in given data.(3)The presented method considers vertex-neighbors coordination tool along with reparametrization to study the interrelationship and ambiguity among objects.
5. Comparison Analysis
In this section, we discuss the comparison of fuzzy type-2 soft graphs with fuzzy soft graphs and type-2 soft graphs.
5.1. Comparison with Fuzzy Soft Graphs
Fuzzy soft graph [21] is a parameterized family of fuzzy graphs, and it is an extension of a soft graph. The fuzzy type-2 soft graph is a parameterized family of VN-fuzzy soft graphs and an extension of type-2 soft graph. Fuzzy type-2 soft graphs show vertex-neighbors coordination relation among objects in a parameterized VN-fuzzy graph. The proposed models take the set of parameters from a given fuzzy vertex set and, corresponding to each selected parameter, there exists a VN-fuzzy soft graph. As fuzzy soft graph is a parameterized family of fuzzy graphs and, corresponding to each parameter, there exists a fuzzy graph. For handling vagueness and ambiguity in decision-making problems, different fuzzy models were introduced. Fuzzy type-2 soft graph shows vertex-neighbors correspondence among objects as well as relations of parameters, while fuzzy soft graphs cannot study these correspondences and thus cannot give accurate and effective results. The decision-making problem discussed in Section 4.1 can be discussed using fuzzy soft graphs.
We consider a fuzzy soft graph , where is a fuzzy soft set over which describes the membership of the objects based upon the given parameters and ; is a fuzzy soft set over describing the membership between two objects corresponding to the given parameters and . A fuzzy soft graph is given in Tables 6 and 7.
The fuzzy graphs and of fuzzy soft graph corresponding to the parameters “carbohydrates” and “protein” are shown in Figure 30.
The fuzzy graphs and and the choice values for all are given in Tables 8 and 9, respectively.
The decision value is from the choice value of fuzzy graph for . Clearly, the dominant object is or . The suitable object determined by fuzzy soft graph as above and fuzzy type-2 soft graph in Section 4.1 is dependent on information determined by selected set of parameters and fuzzy values in VN-fuzzy graphs, respectively. As the coordination among objects varies, the solution changes accordingly. So, in this case, when the objects show close vertex-neighbors coordination according to observed data, fuzzy type-2 soft graph model can be used and in the case when fuzzy relations are given along with different parameters, fuzzy soft graph model can be used.
5.2. Comparison with Type-2 Soft Graphs
In structure of a graph, the vertex-neighbors correspondence has an important role. The type-2 soft graph [32] is based on the correspondence of initial parameters (vertex soft set) and underlying parameters. The type-2 soft graph is an efficient model for dealing with uncertainty occurring in vertex-neighbors’ structure and is applicable in computational analysis, applied intelligence, and decision-making problems. The theory of fuzzy sets has played an important role to form useful models for handling partial membership of objects. To overcome the parameterized limitations of fuzzy set, the theory of fuzzy type-2 soft set was introduced. Fuzzy type-2 soft graph model is a more efficient model as compared to type-2 soft graph model to represent the parametric uncertainty in graphical networks. It is observed that, for the selection of dominant food components in chemical digestion using given type-2 soft information, we are not able to identify any object (dominating component). In this case, the simple type-2 soft information provides no solution. To determine the solution of the problem, it is necessary to have fuzzy information or define a fuzzy relation in order to attain a suitable approximation approach for selecting at least one object. So, fuzzy type-2 soft graph is more reliable in such decision-making problems.
6. Conclusions and Future Directions
Molodtsov’s soft set theory is an effective and rational approach to understand uncertainties in terms of parameters. Type-2 soft sets have been introduced by adding the primary relations among parameters in soft sets. We have introduced the notions of fuzzy type-2 soft sets and fuzzy type-2 soft graphs to study the partial membership and uncertainty of objects along with underlying and primary set of parameters. We have discussed certain properties of fuzzy type-2 soft graphs, regular fuzzy type-2 soft graphs, irregular fuzzy type-2 soft graphs, fuzzy type-2 soft trees, and fuzzy type-2 soft cycles. We have discussed different methods of construction of fuzzy type-2 soft graphs with certain operations and elaborated these concepts with numerical examples. We have studied the importance of fuzzy type-2 soft graphs in chemical digestion and national engineering services. The present study can be extended to various directions including (1) Pythagorean fuzzy type-2 soft graphs, (2) spherical fuzzy type-2 soft graphs, and (3) picture fuzzy type-2 soft trees.
Data Availability
No data were used to support this study.
Conflicts of Interest
The authors declare that they have no conflicts of interest.