# Decision-Making Approach with Fuzzy Type-2 Soft Graphs

Authors: Sundas Shahzadi; Musavarah Sarwar; Muhammad Akram
Journal: Journal of Mathematics
 (2020)

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## 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.


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## Body

## 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 ξ,M, where ξ is a mapping given by ξ:M⟶PE. 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 β−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)J is defined as a pair G=J,K, where K⊆J×J is named as set of edges. Crisp graph J,K is a special case of the fuzzy graph with each vertex and edge of J,K having degree of membership 1. A soft graph corresponding to a crisp graph G is a parameterized family of subgraphs of G. A soft graph on a nonempty set J is a 3−tuple J,K,A such that, for each e∈A, Je,Ke is a graph, where Je⊆J and Ke⊆Je×Je.Definition 1 (see [31]).
LetE,M be a soft universe and let ηE be the set of all T1SSs over E,M. Then a mapping W:B⟶ηE,B⊆M is called a type-2 soft set (T2SS) over E,M and it is denoted by W∗,B. For all δ∈B,W∗δ is a T1SS Wδ,Fδ such that W∗δ=Wδ,Fδ, where Wδ:Fδ⟶PE and Fδ⊂M. We refer to the parameter set B as the “primary set of parameters” although the collection of parameters denoted by ∪Fδ is called “underlying set of parameters.”Definition 2 (see [32]).
Suppose thatG=J,K is a simple graph. Suppose that B⊂J and ΓJ is the set of all T1SSs over J. Suppose that ξ∗,B is a T2SS over J. Then a mapping ξ∗:B⟶ΓJ is said to be a T2SS over J and is denoted as ξ∗,B. For every vertex x∈B, ξ∗,B is a T1SS, where ξ∗x=ξx,Nℬx and ξx:Nℬx⟶PJ can be explained as ξxu=v∈J|uRv∀u∈Nℬx⊆J, This T2SS is said to be a vertex-neighbors type-2 soft set (VN-T2SS) over J.Definition 3 (see [32]).
Suppose thatG=J,K is a simple graph. Suppose that B⊂J and ΓK is the set of all T1SSs over K. Suppose that ξ∗,B is a VN-T2SS over J. Then a mapping ψ∗:B⟶ΓK is said to be a T2SS over K and is denoted as ψ∗,B. For every vertex x∈B, ψ∗,B is a T1SS, where ψ∗x=ψx,NBx and ψx:Nℬx⟶PK can be explained as ψxu=vw∈K|v,w⊆ξxu∀x∈Nℬx⊆J. This T2SS is said to be a VN-T2SS over K.
We present the notations that are used in this research article in Table1.Table 1
List of abbreviations.
AbbreviationDescriptionT1SSType-1 soft setT2SSType-2 soft setVN-T2SSVertex-neighbors type-2 soft setFT1SSFuzzy type-1 soft setFT2SSFuzzy type-2 soft setVN-FT2SSVertex-neighbors fuzzy type-2 soft setFT1SGFuzzy type-1 soft graphFT2SGFuzzy type-2 soft graphFT1STFuzzy type-1 soft treeFT2STFuzzy type-2 soft treeFT2SSTFuzzy type-1 soft subtreeFT2SSTFuzzy type-2 soft subtreeFT1SCFuzzy type-1 soft cycleFT2SCFuzzy type-2 soft cycle


## 3. Fuzzy Type-2 Soft Graphs

We refer to Maji’s [11] fuzzy soft set as fuzzy type-1 soft set (FT1SS). Consider B as a set of parameters that have a random nature (characterization of object, some functions, numeric values, etc.). Consider E as a universal set and the class of all FT1SSs over E will be indicated by PE. 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 E as a universal set and M as the set of parameters. Fuzzy type-2 soft set is defined as follows.Definition 4.
LetE,M be a fuzzy soft universe and let PE be the collection of all FT1SSs over E,M. Then a mapping S∗:B⟶PE, B⊂M, is called a fuzzy type-2 soft set (FT2SS) over E,M and it is denoted by S∗,B. In this case, corresponding to each parameter e∈B, S∗e is FT1SS. Thus, for each e∈B, there exists a FT1SS Se,Le such that S∗e=Se,Le, where Se:Le⟶PE and Le⊂M. In this case, we refer to the parameter set B as the “primary set of parameters,” while the set of parameters ∪Le is known as the “underlying set of parameters.”Definition 5.
LetG=J,K be a fuzzy graph. The set of neighbors of an element j,μj is denoted by Nℬj and defined by Nℬj=i,μi|ij∈K. Then NℬB=∪j∈BNℬj.Definition 6.
LetG=J,K be a fuzzy graph. Suppose that B⊂J and ΓJ is the set of all FT1SSs over J. Suppose that ξ,B is a FT2SS over J. Then a mapping ξ:B⟶ΓJ is said to be a FT2SS over J and is denoted as ξ,B. For every vertex j∈B, ξj=ξj,Nℬj is a FT1SS and ξj:Nℬj⟶PJ can be explained as ξju=v∈J|uRv∀u∈Nℬj⊆J. This FT2SS is said to be a vertex-neighbors fuzzy type-2 soft set (VN-FT2SS) over J.Definition 7.
LetG=J,K be a fuzzy graph. Suppose that B⊂J and ΓK is the set of all T1SSs over K. Suppose that ψ,B is a FT2SS over K. Then a mapping ψ:B⟶ΓK is said to be a FT2SS over K and is denoted as ψ,B. For every vertex j∈B, ψj=ψj,Nℬj is a FT1SS and ψj:Nℬj⟶PK can be explained as ψju=uv∈K|u,v⊆ξju∀u∈Nℬj⊆J. This FT2SS is said to be a VN-FT2SS over K.
ξj=ξj,Nℬj and ψj=ψj,Nℬj∀j∈B are FT1SS over J and K, respectively. If ξju,ψju∀u∈Nℬj represent a fuzzy graph in fuzzy type-2 soft graph G, then ξj,ψj∀j∈B is called FT1SG.Definition 8.
A 5-tupleG=G,ξ,ψ,B,Nℬb is called a fuzzy type-2 soft graph (FT2SG) if it satisfies the following conditions:(a)
G=J,K⊆J×J is a fuzzy graph.(b)
B is a nonempty set of parameters.(c)
ξ,B is a VN-FT2SS over J.(d)
ψ,B is a VN-FT2SS over K.(e)
FT1SS corresponding toξj,ψj∀j∈B represents a VN-fuzzy type-1 soft graph (FT1SG).
A FT2SG can also be defined byG=<ξ,ψ,B>=Zj|j∈B, where Zj=Zj,Nℬj such that Zju  =  ξju,ψjufor allu∈Nℬj.Example 1.
LetG=J,K be a fuzzy graph as shown in Figure 1. Let B=e3,0.5,Nℬe3=e1,0.8,e4,0.4. Suppose that ξ,B and ψ,B are two FT2SSs over J and K, respectively. We have(1)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B.
Defineξe3u=v∈J|uℛv⟺S∞P=0.3 and ψe3u=vw∈K|v,w⊆ξe3u∀u∈Nℬe3⊆J. Then FT2SSs ξ,B and ψ,B are defined as follows:(2)ξe3=e1,0.8,e4,0.4,e4,0.4,e1,0.8,e2,0.7,e3,0.5,e5,0.9,ψe3=e1,0.8,∅,e4,0.4,e1e3,0.4,e1e2,0.6.
Fuzzy type-2 soft graphG=Ze3 is shown in Figure 2.Figure 1
Fuzzy graphG=J,K.Figure 2
Fuzzy type-2 soft graphG=Ze3.Definition 9.
LetG=G,ξ,ψ,B,NℬB be a fuzzy type-2 soft graph; the complement of G is denoted by Gc and defined by Gc=Zcz1,Zcz2,…,Zcznfor  allz1,z2,…,zn∈B, where Zczi=ξczi,ψczi is the complement of FT1SG corresponding to Zzi=ξzi,ψzi for all zi∈B,i=1,2,…,n.Example 2.
LetG=J,K be a fuzzy graph as shown in Figure 3.
LetB=e1,0.7,e5,0.5,Nℬe1=e2,0.8,e3,0.5,e6,0.6, and Nℬe5=e4,0.8,e6,0.6.
Letξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(3)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B.
Define(4)ξe1u=v∈J|uℛv⟺du,v≤0.2,ψe1u=vw∈K|v,w⊆ξe1u,for allu∈Nℬe1⊆J,ξe5u=v∈J|uℛv⟺du,v≤0.2,ψe5u=vw∈K|v,w⊆ξe5u,for allu∈Nℬe5⊆J.
The FT2SSsξ,B and ψ,B are defined as follows:(5)ξe1=e2,0.8,e1,0.7,e3,0.5,e4,0.8,e5,0.5,e3,0.5,e1,0.7,e2,0.8,e6,0.6,e7,0.3,e6,0.6,e3,0.5,e5,0.5,e7,0.3,ψe1=e2,0.8,e1e3,0.1,e4e5,0.1,e3,0.5,e1e2,0.2,e1e6,0.5,e6e7,0.1,e6,0.6,e3e7,0.1,ξe5=e4,0.8,e2,0.8,e5,0.5,e7,0.3,e6,0.6,e7,0.3,e3,0.5,e5,0.5,ψe5=e6,0.6,e3e7,0.1,e4,0.8,∅.
Thenξc,B and ψc,B are defined as follows:(6)ξe1c=e2,0.8,e4,0.8,e3,0.5,e1,0.7,e5,0.5,e3,0.5,e1,0.7,e2,0.8,e6,0.6e7,0.3,e6,0.6,e5,0.5,e3,0.5,e7,0.3,ψe1c=e2,0.8,e1e4,0.7,e1e3,0.4,e1e5,0.5e4e3,0.5,e3e5,0.5,e4e5,0.4,e3,0.5,e1e2,0.5,e1e7,0.3,e1e6,0.1,e2e6,0.6,e2e7,0.3,e6e7,0.2,e6,0.6,e5e7,0.3,e5e3,0.5,e7e3,0.2,ξe5c=e4,0.8,e2,0.8,e5,0.5,e7,0.3,e6,0.6,e7,0.3,e3,0.5,e5,0.5,ψe5c=e4,0.8,e2e7,0.3,e5e7,0.3,e2e5,0.5,e6,0.6,e7e3,0.2,e5e3,0.5,e5e7,0.3.
The complement ofG=Ze1,Ze5 is a FT2SG Gc=Ze1c,Ze5c such that Ze1c=ξe1c,ψe1c is the complement of FT1SG corresponding to Ze1=ξe1,ψe1 and Ze5c=ξe5c,ψe5c is the complement of FT1SG corresponding to Ze5=ξe5,ψe5 as shown in Figure 4.Figure 3
Fuzzy graphG=J,K.Figure 4
Gc=Zce1,Zce5.Definition 10.
LetG be a FT2SG; G is said to be a regular FT2SG if FT1SG corresponding to Zχ is a regular FT1SG for allχ∈B.Proposition 1.
IfG is a regular FT2SG, then Gc is a regular FT2SG.Proof.
LetG be a regular FT2SG. Suppose that Zσ,Nℬσ is a FT1SG corresponding to Zσfor allσ∈B; then Zσjfor allj∈Nℬσ must be a regular fuzzy graph. As we know that complement of a regular graph is regular, Zσcj∀j∈Nℬσ is also a regular fuzzy graph. It provides FT1SG corresponding to a Zcσ for all σ∈B being regular FT1SG. Thus, Gc is a regular FT2SG of G.Definition 11.
LetG be a FT2SG; G is said to be an irregular FT2SG if FT1SG corresponding to Zχ is an irregular FT1SG for allχ∈B.Example 3.
LetG=J,K be a fuzzy graph as shown in Figure 5. Let B=e4,0.4,e5,0.5,Nℬe4=e1,0.1,e3,0.3,e5,0.5, and Nℬe5=e4,0.4,e6,0.9.
Letξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(7)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B.
Define(8)ξe4u=v∈J|uℛv⟺du,v≤0.2,ψe4u=vw∈K|v,w⊆ξe4u,for allu∈Nℬe4⊆J,ξe5u=v∈J|uℛv⟺du,v≤0.2,ψe5u=vw∈K|v,w⊆ξe5u,for allu∈Nℬe5⊆J.
The FT2SSsξ,B and ψ,B are defined as follows:(9)ξe4=e1,0.1,e2,0.2,e3,0.3,e4,0.4,e5,0.5,e3,0.3,e1,0.1,e2,0.2,e4,0.4,e5,0.5,e1,0.1,e4,0.4,e6,0.9,ψe4=e1,0.1,e2e3,0.1,e3e4,0.2,e4e5,0.1,e3,0.3,e1e4,0.1,e1e2,0.1,e5,0.5,e1e4,0.1,ξe5=e4,0.4,e1,0.1,e2,0.2,e3,0.3,e5,0.5,e6,0.9,e6,0.9,e4,0.4,e5,0.5,ψe5=e4,0.4,e1e2,0.1,e2e3,0.1,e5e6,0.1,e6e3,0.3,e6,0.9,e4e5,0.1.
ThenG=Ze4,Ze5 is an irregular FT2SG as shown in Figure 6.Figure 5
Fuzzy graphG=J,K.Figure 6
G=Ze4,Ze5.Proposition 2.
IfG is a regular fuzzy graph, then every FT2SG of G is not necessarily a regular FT2SG.Definition 12.
LetG be a FT2SG; G is called a neighborly irregular FT2SG if FT1SGs corresponding to Zχ are neighborly irregular FT1SG for allχ∈B.Example 4.
LetG=J,K be a fuzzy graph as shown in Figure 7. Let B=e3,0.9,e5,0.4,Nℬe3=e1,0.2,e2,0.4,e4,0.6, and Nℬe5=e4,0.6,e6,0.1,e7,0.3.
Letξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(10)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B.
Define(11)ξe3u=v∈J|uℛv⟺du,v≤0.3,ψe3u=vw∈K|v,w⊆ξe3u,for allu∈Nℬe3⊆J,ξe5u=v∈J|uℛv⟺du,v≤0.4,ψe5u=vw∈K|v,w⊆ξe5u,for allu∈Nℬe5⊆J.
The FT2SSsξ,B and ψ,B are defined as follows:(12)ξe3=e1,0.2,e2,0.4,e3,0.9,e4,0.6,e2,0.4,e1,0.2,e3,0.9,e4,0.6,e4,0.6,e1,0.2,e2,0.4,e3,0.9,e5,0.4,ψe3=e1,0.2,e2e3,0.3,e3e4,0.4,e2,0.4,e1e3,0.2,e3e4,0.4,e1e4,0.1,e4,0.6,e1e2,0.1,e3e2,0.3,e1e3,0.2,ξe5=e4,0.6,e1,0.2,e2,0.4,e3,0.9,e5,0.4,e6,0.1,e5,0.4,e8,0.9,e7,0.3,e7,0.3,e6,0.1,e8,0.9,e5,0.4,ψe5=e4,0.6,e1e2,0.1,e2e3,0.3,e1e3,0.2,e6,0.1,e5e7,0.2,e7e8,0.1,e7,0.3,e6e5,0.2,e6e8,0.1.
ThenG=Ze3,Ze5 is a neighborly irregular FT2SG as shown in Figure 8.Figure 7
Fuzzy graphG=J,K.Figure 8
G=Ze3,Ze5.Definition 13.
LetG be a FT2SG and Zχ=Zχ,Nℬχ is a FT1SG for all χ∈B. An edge uv in G is said to be a FT2S bridge if its removal disconnects Zχu for all u∈Nℬχ.Definition 14.
LetG be a FT2SG and Zχ=Zχ,Nℬχ is a FT1SG for all χ∈B. A vertex z in G is said to be a FT2S cut-vertex if its removal disconnects Zχu for all u∈Nℬχ.Definition 15.
LetG=G,ξ,ψ,B,NℬB be a FT2SG; G is called a fuzzy type-2 soft tree (FT2ST) if FT1SGs corresponding to Zχ are FT1STs for all χ∈B.Example 5.
LetG=J,K be a fuzzy graph as shown in Figure 9. Let B=c,0.6,f,0.2,Nℬc=b,0.2,d,0.4, and Nℬf=e,0.5,g,0.8.
Letξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(13)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B.
Define(14)ξcu=v∈J|uℛv⟺du,v=radG,ψcu=vw∈K|v,w⊆ξcu,for allu∈Nℬc⊆J,ξfu=v∈J|uℛv⟺du,v=radG,ψfu=vw∈K|v,w⊆ξfu,for allu∈Nℬf⊆J.
The FT2SSsξ,B and ψ,B are defined as follows:(15)ξc=b,0.2,g,0.8,f,0.2,d,0.4,a,0.3,g,0.8,ψc=b,0.2,gf,0.2,d,0.4,ag,0.2,ξf=e,0.5,∅,g,0.8,b,0.2,d,0.4,ψf=e,0.5,∅,g,0.8,bd,0.2.
ThenG=Zc,Zf is a FT2ST as shown in Figure 10. It can also be defined as VN-type-2 soft tree.Figure 9
Fuzzy graphG=J,K.Figure 10
G=Zc,Zf.Theorem 1.
LetG be a FT2SG and Zχ=Zχ,Nℬχ is a FT1SG for all χ∈B. If Zχj∀j∈Nℬχ is a FT1SG with n≥3 vertices, then G will not be a complete FT2SG.Proof.
LetG be a FT2SG and Zχ=Zχ,Nℬχ is a FT1SG for all χ∈B. On the contrary, assume that G is a complete FT2SG; then each Zχj∀j∈Nℬχ will also be complete. Let v,w be arbitrary nodes of Zχj joined by a line vw. Since Zχj having n≥3 vertices of G is a FT1SG, then a minimum one vertex η which is connected to v by an edge vη and to w by an edge wη as Zχj be a complete fuzzy graph. Then there is a cycle vwηv. Therefore, Zχj∀j∈Nℬχ cannot be a FT1ST, which is opposite to the fact that Zχj is a connected FT1SG of FT2SG. So, G is not a complete FT2SG.Definition 16.
LetG be a FT2SG and Zχ=Zχ,Nℬχ is a FT1SG for all χ∈B. Then G is called a fuzzy type-2 soft forest if Zχj consists of several disjoined fuzzy trees ∀j∈Nℬχ.Definition 17.
LetG be a FT2SG; G is said to be a FT2SC if FT1SG corresponding to Zχ is a fuzzy type-1 soft cycle, for all χ∈B.Example 6.
LetG=J,K be a fuzzy graph as shown in Figure 11, where(16)J=e1,0.3,e2,0.2,e3,0.2,e4,0.4,e5,0.6,e6,0.2,e7,0.3,e8,0.3,e9,0.1,e10,0.1,K=e1e2,0.1,e2e3,0.1,e3e1,0.1,e3e4,0.1,e5e3,0.1,e5e6,0.1,e6e4,0.1,e7e6,0.3,e7e9,0.1,e7e8,0.3,e9e10,0.1,e8e10,0.1.
LetB=e8,0.3,e9,0.1,Nℬe9=e7,0.3,e10,0.1, and Nℬe8=e7,0.3,e10,0.1.
Letξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(17)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B.
Define(18)ξe9u=v∈J|uℛv⟺0.5≤du,v≤0.7,ψe9u=vw∈K|v,w⊆ξe9u,for allu∈Nℬe9⊆J,ξe8u=v∈J|uℛv⟺0.5≤du,v≤0.7,ψe8u=vw∈K|v,w⊆ξe8u,for allu∈Nℬe8⊆J.
The FT2SSsξ,B and ψ,B are defined as follows:(19)ξe9=e7,0.3,e1,0.3,e2,0.2,e3,0.2,e10,0.1,e3,0.2,e4,0.4,e5,0.6,e6,0.2,ψe9=e7,0.3,e3e2,0.1,e1e3,0.1,e1e2,0.1,e10,0.1,e4e3,0.1,e3e5,0.1,e6e5,0.1,e6e4,0.1,ξe8=e7,0.3,e1,0.3,e2,0.2,e3,0.2,e10,0.1,e3,0.2,e4,0.4,e5,0.6,e6,0.2,ψe8=e7,0.3,e3e2,0.1,e1e3,0.1,e1e2,0.1,e10,0.1,e4e3,0.1,e3e5,0.1,e6e5,0.1,e6e4,0.1.
We can check thatG=Ze9,Ze8 is a FT2SC as shown in Figure 12. It is also defined as a fuzzy VN-type-2 soft cycle.Figure 11
Fuzzy graphG=J,K.Figure 12
G=Ze9,Ze8.Example 7.
LetG=J,K be a fuzzy graph as shown in Figure 13. Let B=a,0.4,b,0.2⊂J, Nℬa=b,0.2,c,0.3,d,0.3, and Nℬb=a,0.4,c,0.3,d,0.3. Let ξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(20)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B.
Defineξau=v∈J|uℛv⟺du,v≤0.2,ψau=vw∈K|v,w⊆ξau∀u∈Nℬa⊆J,ξbu=v∈J|uℛv⟺du,v≤0.2,ψbu=vw∈K|v,w⊆ξbu∀u∈Nℬb⊆J.
The FT2SSsξ,B and ψ,B are defined as follows:(21)ξa=b,0.2,a,0.4,c,0.3,d,0.3,c,0.3,a,0.4,b,0.2,d,0.3,e,0.4,d,0.3,a,0.4,b,0.2,c,0.3,e,0.4,ψa=b,0.2,ad,0.1,ac,0.2,c,0.3,ab,0.2,ad,0.1,bd,0.1,de,0.2,d,0.3,ab,0.2,ac,0.2,bc,0.1,ce,0.2,ξb=a,0.4,c,0.3,b,0.2,d,0.3,c,0.3,a,0.4,b,0.2,e,0.4,d,0.3,d,0.3,a,0.4,b,0.2,c,0.3,e,0.4,ψb=a,0.4,bc,0.1,bd,0.1,c,0.3,ad,0.1,ed,0.2,ab,0.2,bd,0.1,d,0.3,ec,0.2,ab,0.2,ac,0.2,bc,0.1.
Za=ξa,ψa and Zb=ξb,ψb are FT1SGs as shown in Figure 14. We can see that Zab=ξab,ψab,Zac=ξac,ψac,Zad=ξad,ψad,Zbc=ξbc,ψbc and Zbd=ξbd,ψbd are all not trees. Hence G=Za,Zb is not a FT2ST and G is also not a FT2SC.Figure 13
Fuzzy graphG=J,K.Figure 14
G=Zb,Za.Proposition 3.
Every fuzzy type-2 soft cycle is a regular fuzzy type-2 soft cycle.Proof.
LetG be a FT2SC. Let Zχ,Nℬχ be a T1FSC corresponding to Zχ for every χ∈B. Then, Zχj is a cycle ∀j∈Nℬχ. We know that cycle is a path that is closed and every vertex of cycle is of degree 2; this signifies that Zχj is a regular fuzzy graph for all j∈Nℬχ. Therefore, Zχ,Nℬχ is a regular FT1SG, for all χ∈B. Hence G is a regular FT2SG.Definition 18.
LetG1=<ξ1,ψ1,B1> and G2=<ξ2,ψ2,B2> be two FT2STs. G2 is a fuzzy type-2 soft subtree (FT2SST) of G1 if(i)
B2⊆B1(ii)
For eachj∈B2, FT1ST corresponding to Z2j=ξ2j,ψ2j is a fuzzy type-1 soft subtree (FT1SST) of FT1ST corresponding to Z1j=ξ1j,ψ1jExample 8.
LetG=J,K be a fuzzy graph as shown in Figure 15, where(22)J=e1,0.3,e2,0.2,e3,0.2,e4,0.4,e5,0.6,e6,0.2,e7,0.3,K=e1e2,0.1,e2e3,0.1,e3e4,0.2,e5e4,0.1,e5e6,0.1,e7e6,0.2,e7e1,0.1.
LetB=e2,0.2,e4,0.4,B∗=e2,0.2,e4,0.4,Nℬe2=e1,0.3,e3,0.2,Nℬe4=e3,0.2,e5,0.6. Let ξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(23)ξj=ξj,Nℬjψj=ψj,Nℬj,for allj∈B.
Defineξe2u=v∈J|uℛv⟺du,v≤0.2,ψe2u=vw∈K|v,w⊆ξe2u∀u∈Nℬe2⊆J and ξe4u=v∈J|uℛv⟺du,v=0.2,ψe4u=vw∈K|v,w⊆ξe4u∀u∈Nℬe4⊆J.
Then FT2SSsξ,B and ψ,B are defined as follows:(24)ξe2=e1,0.3,e2,0.2,e3,0.2,e7,0.3,e3,0.2,e1,0.3,e2,0.2,e4,0.4,ψe2=e1,0.3,e2e3,0.1,e3,0.2,e1e2,0.1,ξe4=e3,0.2,e1,0.3,e4,0.4,e5,0.6,∅,ψe4=e3,0.2,∅,e5,0.6,∅.
G=Ze2,Ze4 is a FT2ST as shown in Figure 16.
Letξ′,N and ψ′,N be two FT2SSs over J and K, respectively. We have(25)ξ′j=ξj,Nℬj,ψ′j=ψj,Nℬj,for allj∈N.
Defineξe2′u=v∈J|uℛv⟺du,v≤0.3,ψe2′u=vw∈K|v,w⊆ξe2′u∀u∈Nℬe2⊆J and ξe4′u=v∈J|uℛv⟺du,v≤0.3,ψe4′u=vw∈K|v,w⊆ξe4′u∀u∈Nℬe4⊆J.
FT2SSsξ′,N and ψ′,N are defined as follows:(26)ξe2′=e1,0.3,e2,0.2,e3,0.2,e7,0.3,e3,0.2,e1,0.3,e2,0.2,e4,0.4,e7,0.3,ψe2′=e1,0.3,e2e3,0.1,e3,0.2,e1e7,0.1,e1e2,0.1,ξe4′=e3,0.2,e1,0.3,e2,0.2,e4,0.4,e7,0.3,e5,0.6,e4,0.4,e6,0.2,ψe4′=e3,0.2,e2e1,0.1e1e7,0.1,e5,0.6,∅.
G′=Z′e2,Z′e4 is a FT2SST of G as shown in Figure 17. We can see that B⊂B∗ and Ze2⊆Z′e2,Ze4⊆Z′e4. Hence, G is a FT2SST of G′.Figure 15
G=J,K is a fuzzy graph.Figure 16
G=Ze2,Ze4.Figure 17
G′=Ze4′,Ze2′.Theorem 2.
LetG1=<ξ1,ψ1,B1> and G2=<ξ2,ψ2,B2> be two FT2STs. Then G2 is said to be a FT2SST of G1 if and only if ξ2⊆ξ1 and ψ2⊆ψ1.Proof.
LetG2 be a FT2SST of G1. Then, by using the definition of FT2SST,(i)
B2⊆B1(ii)
For allj∈B2, FT1ST corresponding to Z2j=ξ2j,ψ2j is a FT1SST of FT1ST corresponding to Z1j=ξ1j,ψ1j
Since FT1ST corresponding toZ2j is a FT1SST of FT1ST corresponding to Z1j for all j∈B2, we have ξ2⊆ξ1 and ψ2⊆ψ1∀j∈B2. Conversely, we have ξ2j⊆ξ1j and ψ2j⊆ψ1j∀j∈B2. As G1 is a fuzzy type-2 soft tree, fuzzy type-1 soft set corresponding to Z1j forms a FT1ST of G2 for all j∈B1. Also, G2 is a fuzzy type-2 soft tree, and fuzzy type-1 soft set corresponding to Z2j forms a FT1ST of G1 for all j∈B2. This implies that FT1ST corresponding to Z2j is a FT1SST of FT1ST corresponding to Z1j for all j∈B2. Hence, G2 is a FT2SST of G1.Definition 19.
LetG1=<ξ1,ψ1,B1> and G2=<ξ2,ψ2,B2> be two FT2STs. The union of G1 and G2 is denoted by G1∪G2=G=<ξ,ψ,P>, where P=B1∪B2, such that(27)ξv=ξ1v,ifv∈B1−B2,ξ2v,ifv∈B2−B1,ξ1v∪ξ2v,ifv∈B1∩B2,ψv=ψ1v,ifv∈B1−B2,ψ2v,ifv∈B2−B1,ψ1v∪ψ2v,ifv∈B2∩B1,where ξ1v∪ξ2v,ψ1v∪ψ2v for all v∈B1∩B2 relates to the fuzzy type-1 soft union between the relevant FT1STs corresponding to ξ1v,ψ1v and ξ2v,ψ2v, respectively. It can be written as G1∪G2=Zv=ξv,ψv|v∈P.Theorem 3.
LetG1=<ξ1,ψ1,B1> and G2=<ξ2,ψ2,B2> be two FT2STs with B1∩B2=∅. Then G1∪G2 is a FT2ST.Proof.
LetG1=<ξ1,ψ1,B1> and G2=<ξ2,ψ2,B2> be two FT2STs. The union of G1 and G2 is denoted by G1∪G2=G=<ξ,ψ,P>, where P=B1∪B2 is defined ∀v∈P:(28)ξv=ξ1v,ifv∈B1−B2,ξ2v,ifv∈B2−B1,ξ1v∪ψ2v,ifv∈B1∩B2,ψv=ψ1v,ifv∈B1−B2,ψ2v,ifv∈B2−B1,ψ1v∪ψ2v,ifv∈B2∩B1,where ψ1v∪ψ2v for all v∈B2∩B1 relates to the fuzzy type-1 soft extended union among the relevant FT1STs corresponding to ψ1v and ψ2v, respectively, and ξ1v∪ξ2v for all v∈B1∩B2 relates to the fuzzy type-1 soft union between the relevant FT1STs corresponding to ξ1v and ξ2v, respectively. Since G1 is a FT2ST, FT1ST corresponding to ξ1jandψ1j is a FT2ST for all j∈B1−B2.
SinceG2 is a FT2ST, FT1ST corresponding to ξ2jandψ2j is a FT2ST for all j∈B2−B1. It is given that B1∩B2=∅. Thus, G1∪G2=G=<ξ,ψ,B1∩B2> is a FT2ST.Definition 20.
LetG1=<ξ1,ψ1,B1> and G2=<ξ2,ψ2,B2> be two FT2STs. The intersection of G1 and G2 is denoted by G1∩G2=G=<ξ,ψ,P>, where P=B1∪B2 such that(29)ξv=ξ1v,ifv∈B1−B2,ξ2v,ifv∈B2−B1,ξ1v∩ξ2v,ifv∈B1∩B2,ψv=ψ1v,ifv∈B1−B2,ψ2v,ifv∈B2−B1,ψ1v∩ψ2v,ifv∈B2∩B1,where ξ1v∩ξ2v,ψ1v∩ψ2v for all v∈B1∩B2 relates to the fuzzy type-1 soft intersection between the relevant FT1STs corresponding to ξ1v,ψ1v and ξ2v,ψ2v, respectively.
It can be written asG1∩G2=Zv=ξv,ψv|v∈P.Example 9.
LetG be a fuzzy graph as shown in Figure 18. Let B=a,0.9,b,0.1,B∗=a,0.9,\\e,0.2. It can be written as Nℬa=v,0.7,b,0.1,Nℬb=a,0.9,c,0.1,Nℬe=d,0.3,f,0.5. Let ξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(30)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B.
Defineξau=v∈J|uℛv⟺du,v≤0.3,ψau=vw∈K|v,w⊆ξau∀u∈Nℬa and ξbu=v∈J|uℛv⟺du,v≤0.3,ψbu=vw∈K|v,w⊆ξbu∀z∈Nℬb.
The FT2SSsξ,B and ψ,B are defined as follows:(31)ξa=b,0.1,a,0.9,c,0.1,d,0.3,v,0.7,d,0.3,g,0.1,f,0.5,h,0.2,ψa=b,0.1,cd,0.1,v,0.7,gh,0.1,gf,0.1,ξb=c,0.1,a,0.9,b,0.1,d,0.3,e,0.2,a,0.9,b,0.1,c,0.1,d,0.3,ψb=c,0.1,ab,0.1,ed,0.2,a,0.9,bc,0.1,cd,0.1.
Letξ′,B∗ and ψ′,B∗ be two FT2SSs over J and K, respectively. We have(32)ξ′j=ξj,Nℬj,ψ′j=ψj,Nℬj,for allj∈B∗,defineξa′u=v∈J|uℛv⟺du,v≤0.2,ψa′u=vw∈K|v,w⊆ξa′u,∀u∈Nℬa⊆J,ξe′u=v∈J|uℛv⟺du,v≤0.3,ψe′u=vw∈K|v,w⊆ξe′u,∀u∈Nℬe⊆J.
The FT2SSsξ′,B∗ and ψ′,B∗ are defined as follows:(33)ξa′=b,0.1,a,0.9,c,0.1,d,0.3,v,0.7,g,0.1,h,0.2,ψa′=b,0.1,cd,0.1,v,0.7,gh,0.1,ξe′=d,0.3,a,0.9,b,0.1,c,0.1,e,0.2,v,0.7,f,0.5,e,0.2,g,0.1,h,0.2,v,0.7,ψe′=d,0.3,bc,0.1,av,0.5,ba,0.1,f,0.5,vh,0.1,hg,0.1.
ThenG=Za,Zb and G′=Z′e,Z′a are FT2STs as shown in Figure 19. By the definition of intersection of FT2STs, ξa=ξa∩ξ′a and ψa=ψa∩ψ′awherea∈B∗∩B.
Therefore,G1∩G2=G=<ξ,ψ,B∪B∗> is a FT2ST as shown in Figure 20.Figure 18
Fuzzy graphG=J,K.Figure 19
G=Za,Zb and G′=Z′e,Z′a.Figure 20
Intersection ofG and G′ is G∩G′.Definition 21.
LetG1=<ξ1,ψ1,M1> and G2=<ξ2,ψ2,B2> be two FT2STs. The AND operation of G1 and G2 is denoted by G1∧G2=G=<ξ,ψ,B1×B2> such that ξχ,η=ξ1χ∧ξ2η,ψχ,η=ψ1χ∧ψ2ηforallχ,η∈B1×B2. ξχ,η,ψχ,η for all χ,η∈B1×B2 is the fuzzy type-1 soft AND operation between the relevant FT1SGs corresponding to ξ1χ,ψ1χ and ξ2η,ψ2η, respectively.Example 10.
LetG=J,K be the fuzzy graph as shown in Figure 21, where(34)J=e1,0.3,e2,0.1,e3,0.2,e4,0.4,e5,0.6,e6,0.2,e7,0.3,e8,0.3,K=e1e2,0.1,e2e3,0.1,e3e4,0.2,e5e4,0.3,e5e6,0.1,e7e6,0.2,e7e8,0.2,e8e1,0.1.
LetB=e3,0.2,e4,0.4,B∗=e7,0.3,Nℬe3=e2,0.1,e4,0.4,Nℬe4=e3,0.2,e5,0.6,Nℬe7=e6,0.2,e8,0.3.
Letξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(35)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allJ∈B,=ψj,Nℬj,defineξe3u=v∈J|uℛv⟺du,v≤0.3,ψe3u=vw∈K|v,w⊆ξe3u,∀u∈Nℬe3,ξe4u=v∈J|uℛv⟺du,v≤0.4,ψe4u=vw∈K|v,w⊆ξe4u,∀z∈Nℬe3.
The FT2SSsξ,B and ψ,B are defined as follows:(36)ξe3=e4,0.4,e2,0.1,e3,0.2,e5,0.6,e2,0.1,e1,0.3,e3,0.2,e4,0.4,e8,0.3,ψe3=e4,0.4,e2e3,0.1,e2,0.1,e3e4,0.2,e1e8,0.1,ξe4=e5,0.6,e4,0.4,e6,0.2,e7,0.3,e3,0.2,e1,0.3,e2,0.1,e4,0.4,e8,0.3,ψe4=e3,0.2,e8e1,0.1,e1e2,0.1,e5,0.6,e6e7,0.2.
ThenG=Ze3,Ze4 is a FT2ST. Let ξ′,B∗ and ψ′,B∗ be two FT2SSs over J and K, respectively. We have(37)ξ′j=ξj,Nℬj,ψ′j=ψj,Nℬj,for allj∈B∗.
Defineξe7′u=v∈J|uℛv⟺du,v≤0.4, ψe7′u=vw∈K|v,w⊆ξe7′u∀u∈Nℬe7⊆J,(38)ξe7′=e8,0.3,e1,0.3,e2,0.1,e3,0.2,e6,0.2,e7,0.3,e6,0.2,e4,0.4,e5,0.6,e7,0.3,e8,0.3,ψe7′=e8,0.3,e1e2,0.1,e2e3,0.1,e7e6,0.2,e6,0.2,e4e5,0.3,e7e8,0.2.
G′=Z′e7 is a FT2ST. The AND operation of G and G′ is defined as follows:(39)ξe3,e7=ξe3∧ξe7′e4,0.4,e8,0.3,e3,0.2,e2,0.1,e4,0.4,e6,0.2,e5,0.6,e2,0.1,e6,0.2,e4,0.4,e8,0.3,e8,0.3,e2,0.1,e1,0.3,e3,0.2,ψe3,e7=ψe3∧ψe7′e4,0.4,e8,0.3,e2e3,0.1,e4,0.4,e6,0.2,∅,e2,0.1,e8,0.3,∅,e2,0.1,e6,0.2,∅,ξe4,e7=ξe4∧ξe7′e5,0.6,e8,0.3,e6,0.2,e7,0.3,e5,0.6,e6,0.2,e7,0.3,e4,0.4,e3,0.2,e8,0.3,e1,0.3,e2,0.1,e3,0.2,e6,0.2,e8,0.3,e4,0.4,ψe4,e7=ψe4∧ψe7′e5,0.6,e8,0.3,e6e7,0.2,e5,0.6,e6,0.2,∅,e3,0.2,e8,0.3,e2e1,0.1,e3,0.2,e6,0.2,∅.
The AND operation ofG and G′ is shown in Figure 22.Figure 21
Fuzzy graphG=J,K.Figure 22
AND operation ofG and G′ is G∧G′=Ze3,e7,Ze4,e7.Definition 22.
LetG1=<ξ1,ψ1,M1> and G2=<ξ2,ψ2,B2> be two FT2STs. The OR operation of G1 and G2 is denoted by G1∨G2=G=<ξ,ψ,B1×B2> such that ξχ,η=ξ1χ∨ξ2η,ψχ,η=ψ1χ∨ψ2ηfor allχ,η∈B1×B2. ξχ,η,ψχ,η for all χ,η∈B1×B2 is the fuzzy type-1 soft OR operation between the relevant FT1SGs corresponding to ξ1χ,ψ1χ and ξ2η,ψ2η, 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 Algorithm1 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.Algorithm 1: Algorithm for the selection of most suitable objects.
(1)
Input the fuzzy graphG=J,K.(2)
Input the choice parameter setB=e1,e2,…,en for suitable selection of the object.(3)
Input the VN-FT2SSξ,B over J and VN-FT2SS ψ,B over K.(4)
Construct a FT2SGG=<ξ,ψ,B>=Zj|j∈B, where Zj=Zj,Nℬj such that Zju = ξju,ψju,∀u∈Nℬj.(5)
Construct the resultant VN-fuzzy graph by taking the intersection of vertex-neighbors fuzzy graphsZ∗j=∩uZju,∀u∈Nℬj.(6)
Tabular representation of resultant VN-fuzzy graphZ∗j∀j∈B with the choice values Cij.(7)
The decision isSi if Si=∨in∧jCij.(8)
Ifi has more than one value, then any one of Si may be chosen.Figure 23
Flow chart for suitable selection of objects.


### 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 carbohydrates0.9g, protein (0.7g), lipids 0.1g, nucleic acids 0.3g, involuntary signal (0.3), pepsin (0.2), amylase (0.2), and trypsin (0.1), represented as vertices donated by e1,e2,e3,e4,e5,e6,e7,e8, 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 pancreasNote 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(40)J=e1,0.9,e2,0.7,e3,0.1,e4,0.3,e5,0.3,e6,0.2,e7,0.2,e8,0.1,K=e1e2,0.3,e1e4,0.3,e3e2,0.1,e3e4,0.1,e5e4,0.2,e5e6,0.2,e5e8,0.1,e8e7,0.1,e6e7,0.1.In fuzzy graphJ,K as shown in Figure 24, edges represent the amount of energy utilized by the body in order to carry out the digestion process. Let B=e1,0.9,e2,0.7 represent the amounts of carbohydrates and protein released when 25 g of spinach is consumed. We have Nℬe1=e2,0.7,e4,0.3,Nℬe2=e3,0.1,e1,0.9.Figure 24
G=J,K.Letξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(41)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B,defineξe1u=v∈J|uℛv⟺du,v≤0.5,ψe1u=vw∈K|v,w⊆ξe1u,ξe2u=v∈J|uℛv⟺du,v≤0.5,ψe2u=vw∈K|v,w⊆ξe2u.The FT2SSsξ,B and ψ,B are defined as follows:(42)ξe1=e2,0.7,e1,0.9,e3,0.1,e4,0.3,e5,0.3,e4,0.3,e1,0.9,e2,0.7,e3,0.1,e5,0.3,e6,0.2,e7,0.2,e8,0.1,ψe1=e2,0.7,e3e4,0.1,e1e4,0.3,e5e4,0.2,e4,0.5,e1e2,0.3,e3e2,0.1,e5e6,0.2,e5e8,0.1,e6e7,0.1,e7e8,0.1,ξe2=e1,0.9,e2,0.7,e3,0.1,e4,0.3,e5,0.3,e3,0.1,e1,0.9,e2,0.7,e4,0.3,e5,0.3,e6,0.2,e8,0.1,ψe2=e1,0.9,e5e4,0.2,e3e4,0.2,e2e3,0.1,e3,0.4,e1e2,0.3,e1e4,0.3,e4e5,0.2,e5e6,0.2,e5e8,0.1.The fuzzy type-2 soft graphG is shown in Figure 25.Figure 25
Fuzzy type-2 soft graphG=Ze1,Ze2.The tabular representations of resultant vertex-neighbors fuzzy graphsZ∗ej shown in Figure 26 corresponding to the parameter ej,j=1,2 with the choice values Cij=∑kSik for all i,k are given in Tables 2 and 3.Figure 26
Resultant VN-fuzzy graphsZ∗e1 and Z∗e2.Table 2
The tabular representation ofZ∗e1 with choice values.
e1e2e3e4e5e6e7e8Ci1e100.300.300000.6e20.300.1000000.4e300.100.100000.2e40.300.100.20000.6e50000.200.200.10.5e600000.200.100.3e7000000.100.10.2e800000.100.100.2Table 3
The tabular representation ofZ∗e2 with choice values Ci2.
e1e2e3e4e5e6e7Ci2e100.300.30000.6e20.300.100000.4e300.100.10000.2e40.300.100.2000.6e50000.200.20.10.5e600000.2000.2e800000.1000.1The decision value isSi=∨i7∧jCij=∨i=170.6∧0.6,0.4∧0.4,0.2∧0.2,0.6∧0.5,0.3∧0.2,0.2∧0,0.3∧0.2=0.6 from the choice value Cij of fuzzy type-2 soft graphs for j=1,2. The prominent food components are e1 as carbohydrates and e2 as lipids as carbohydrates are consumed as sugar and lipids are consumed as fats. Clearly, the dominant food components are e1 or e4.


### 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 graphG=J,K as shown in Figure 27, vertices represent some important projects.(43)J=a=Water Supply,0.9,b=Sewerage,0.7,c=Drainage,0.6,d=Solid Waste Management,0.4,e=Plumbing,0.3,f=Industrial Wastes,0.2.Figure 27
Fuzzy graphG=J,K.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 ase,0.3,d,0.4, respectively, and B=e,0.3,d,0.4⊂J. The vertex-neighbors of these selected projects are Nℬe=b,0.7,c,0.6,d,0.4,f,0.2 and Nℬd=a,0.9,c,0.6,e,0.3,f,0.2. Let ξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(44)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B,defineξeu=v∈J|uℛv⟺du,v≤0.2,ψeu=vw∈K|v,w⊆ξeu,∀u∈Nℬe⊆J,ξdu=v∈J|uℛv⟺0.1≤du,v≤0.2,ψdu=vw∈K|v,w⊆ξdu,∀u∈Nℬd⊆J.The FT2SSsξ,B and ψ,B are defined as follows:(45)ξe=f,0.2,a,0.9,b,0.7,d,0.4,e,0.3,d,0.4,f,0.2,c,0.6,e,0.3,b,0.7,e,0.3,f,0.2,ψe=f,0.2,ad,0.3,ab,0.4,de,0.3,eb,0.2,d,0.4,∅,c,0.6,∅,b,0.7,fe,0.1,ξd=e,0.3,b,0.7,c,0.6,f,0.2,a,0.9,f,0.2,f,0.2,a,0.9,b,0.7,d,0.4,e,0.3,c,0.6,e,0.3,ψd=e,0.3,bc,0.4,fb,0.1,a,0.9,∅,c,0.6,∅,f,0.2,ad,0.3,ab,0.4,de,0.3,eb,0.2.FT1SGs corresponding toZe=ξe,ψe and Zd=ξd,ψd, respectively, are shown in fuzzy type-2 soft graph 28. (Figure 28)Figure 28
Fuzzy type-2 soft graph for national engineering services.The tabular representations of resultant vertex-neighbors fuzzy graphsZ∗e and Z∗d shown in Figure 29 with the choice values Cij=∑kSik for all i,k are given in Tables 4 and 5.Figure 29
Resultant VN-fuzzy graphsZ∗e and Z∗d.Table 4
The tabular representation ofZ∗e with choice values.
abdefCi1a00.40.3000.7b0.4000.200.6d0.3000.300.6e00.20.300.10.6f0000.100.1Table 5
The tabular representation ofZ∗d with choice values.
abcdefCi2a00.400.3000.7b0.400.400.20.11.1c00.400000.4d0.30000.300.6e00.200.3000.5f00.100000.1The decision value isSi=∨i6∧jCij=∨i=160.7∧0.7,0.6∧1.1,0∧0.4,0.6∧0.6,0.6∧0.5,0.1∧0.1=0.7, from the choice value Cij of fuzzy type-2 soft graphs for j=1,2. The optimal project is “a= 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.


## 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 carbohydrates0.9g, protein (0.7g), lipids 0.1g, nucleic acids 0.3g, involuntary signal (0.3), pepsin (0.2), amylase (0.2), and trypsin (0.1), represented as vertices donated by e1,e2,e3,e4,e5,e6,e7,e8, 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 pancreasNote 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(40)J=e1,0.9,e2,0.7,e3,0.1,e4,0.3,e5,0.3,e6,0.2,e7,0.2,e8,0.1,K=e1e2,0.3,e1e4,0.3,e3e2,0.1,e3e4,0.1,e5e4,0.2,e5e6,0.2,e5e8,0.1,e8e7,0.1,e6e7,0.1.In fuzzy graphJ,K as shown in Figure 24, edges represent the amount of energy utilized by the body in order to carry out the digestion process. Let B=e1,0.9,e2,0.7 represent the amounts of carbohydrates and protein released when 25 g of spinach is consumed. We have Nℬe1=e2,0.7,e4,0.3,Nℬe2=e3,0.1,e1,0.9.Figure 24
G=J,K.Letξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(41)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B,defineξe1u=v∈J|uℛv⟺du,v≤0.5,ψe1u=vw∈K|v,w⊆ξe1u,ξe2u=v∈J|uℛv⟺du,v≤0.5,ψe2u=vw∈K|v,w⊆ξe2u.The FT2SSsξ,B and ψ,B are defined as follows:(42)ξe1=e2,0.7,e1,0.9,e3,0.1,e4,0.3,e5,0.3,e4,0.3,e1,0.9,e2,0.7,e3,0.1,e5,0.3,e6,0.2,e7,0.2,e8,0.1,ψe1=e2,0.7,e3e4,0.1,e1e4,0.3,e5e4,0.2,e4,0.5,e1e2,0.3,e3e2,0.1,e5e6,0.2,e5e8,0.1,e6e7,0.1,e7e8,0.1,ξe2=e1,0.9,e2,0.7,e3,0.1,e4,0.3,e5,0.3,e3,0.1,e1,0.9,e2,0.7,e4,0.3,e5,0.3,e6,0.2,e8,0.1,ψe2=e1,0.9,e5e4,0.2,e3e4,0.2,e2e3,0.1,e3,0.4,e1e2,0.3,e1e4,0.3,e4e5,0.2,e5e6,0.2,e5e8,0.1.The fuzzy type-2 soft graphG is shown in Figure 25.Figure 25
Fuzzy type-2 soft graphG=Ze1,Ze2.The tabular representations of resultant vertex-neighbors fuzzy graphsZ∗ej shown in Figure 26 corresponding to the parameter ej,j=1,2 with the choice values Cij=∑kSik for all i,k are given in Tables 2 and 3.Figure 26
Resultant VN-fuzzy graphsZ∗e1 and Z∗e2.Table 2
The tabular representation ofZ∗e1 with choice values.
e1e2e3e4e5e6e7e8Ci1e100.300.300000.6e20.300.1000000.4e300.100.100000.2e40.300.100.20000.6e50000.200.200.10.5e600000.200.100.3e7000000.100.10.2e800000.100.100.2Table 3
The tabular representation ofZ∗e2 with choice values Ci2.
e1e2e3e4e5e6e7Ci2e100.300.30000.6e20.300.100000.4e300.100.10000.2e40.300.100.2000.6e50000.200.20.10.5e600000.2000.2e800000.1000.1The decision value isSi=∨i7∧jCij=∨i=170.6∧0.6,0.4∧0.4,0.2∧0.2,0.6∧0.5,0.3∧0.2,0.2∧0,0.3∧0.2=0.6 from the choice value Cij of fuzzy type-2 soft graphs for j=1,2. The prominent food components are e1 as carbohydrates and e2 as lipids as carbohydrates are consumed as sugar and lipids are consumed as fats. Clearly, the dominant food components are e1 or e4.


## 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 graphG=J,K as shown in Figure 27, vertices represent some important projects.(43)J=a=Water Supply,0.9,b=Sewerage,0.7,c=Drainage,0.6,d=Solid Waste Management,0.4,e=Plumbing,0.3,f=Industrial Wastes,0.2.Figure 27
Fuzzy graphG=J,K.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 ase,0.3,d,0.4, respectively, and B=e,0.3,d,0.4⊂J. The vertex-neighbors of these selected projects are Nℬe=b,0.7,c,0.6,d,0.4,f,0.2 and Nℬd=a,0.9,c,0.6,e,0.3,f,0.2. Let ξ,B and ψ,B be two FT2SSs over J and K, respectively. We have(44)ξj=ξj,Nℬj,ψj=ψj,Nℬj,for allj∈B,defineξeu=v∈J|uℛv⟺du,v≤0.2,ψeu=vw∈K|v,w⊆ξeu,∀u∈Nℬe⊆J,ξdu=v∈J|uℛv⟺0.1≤du,v≤0.2,ψdu=vw∈K|v,w⊆ξdu,∀u∈Nℬd⊆J.The FT2SSsξ,B and ψ,B are defined as follows:(45)ξe=f,0.2,a,0.9,b,0.7,d,0.4,e,0.3,d,0.4,f,0.2,c,0.6,e,0.3,b,0.7,e,0.3,f,0.2,ψe=f,0.2,ad,0.3,ab,0.4,de,0.3,eb,0.2,d,0.4,∅,c,0.6,∅,b,0.7,fe,0.1,ξd=e,0.3,b,0.7,c,0.6,f,0.2,a,0.9,f,0.2,f,0.2,a,0.9,b,0.7,d,0.4,e,0.3,c,0.6,e,0.3,ψd=e,0.3,bc,0.4,fb,0.1,a,0.9,∅,c,0.6,∅,f,0.2,ad,0.3,ab,0.4,de,0.3,eb,0.2.FT1SGs corresponding toZe=ξe,ψe and Zd=ξd,ψd, respectively, are shown in fuzzy type-2 soft graph 28. (Figure 28)Figure 28
Fuzzy type-2 soft graph for national engineering services.The tabular representations of resultant vertex-neighbors fuzzy graphsZ∗e and Z∗d shown in Figure 29 with the choice values Cij=∑kSik for all i,k are given in Tables 4 and 5.Figure 29
Resultant VN-fuzzy graphsZ∗e and Z∗d.Table 4
The tabular representation ofZ∗e with choice values.
abdefCi1a00.40.3000.7b0.4000.200.6d0.3000.300.6e00.20.300.10.6f0000.100.1Table 5
The tabular representation ofZ∗d with choice values.
abcdefCi2a00.400.3000.7b0.400.400.20.11.1c00.400000.4d0.30000.300.6e00.200.3000.5f00.100000.1The decision value isSi=∨i6∧jCij=∨i=160.7∧0.7,0.6∧1.1,0∧0.4,0.6∧0.6,0.6∧0.5,0.1∧0.1=0.7, from the choice value Cij of fuzzy type-2 soft graphs for j=1,2. The optimal project is “a= 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 graphG=Φ,Ψ,M, where Φ,M is a fuzzy soft set over V=e1,e2,e3,e4,e5,\\e6,e7,e8 which describes the membership of the objects based upon the given parameters e1 and e2; Ψ,M is a fuzzy soft set over E=e1e2,e1e4,e2e3,e3e4,e4e5,e5e6,e5e6,e5e8,e6e7,e8e7⊆V×V describing the membership between two objects corresponding to the given parameters e1 and e2. A fuzzy soft graph G=He1,He2 is given in Tables 6 and 7.Table 6
Tabular representation of a fuzzy soft vertex set.
Φe1e2e3e4e5e6e7e8e10.80.70.20.40.30.30.20.2e20.80.50.50.40.40.50.60.5Table 7
Tabular representation of a fuzzy soft edge set.
Ψe1e2e1e6e1e4e1e7e2e3e2e4e2e8e3e4e3e7e4e5e5e6e5e8e7e8e10.30.20.00.00.20.10.00.10.10.20.10.10.1e20.30.00.20.10.00.00.10.20.10.20.10.10.0The fuzzy graphsHe1 and He2 of fuzzy soft graph G=He1,He2 corresponding to the parameters “carbohydrates” and “protein” are shown in Figure 30.Figure 30
Fuzzy soft graphG=He1,He2.The fuzzy graphsHe1 and He2 and the choice values Cik=∑jSij for all i,j,k=1,2 are given in Tables 8 and 9, respectively.Table 8
Tabular representation ofHe1 with choice values.
e1e2e3e4e5e6e7e8Ci1e10.00.30.00.00.00.20.00.00.5e20.30.00.20.10.00.00.00.00.6e30.00.20.00.10.00.00.10.00.4e40.00.10.10.00.20.00.00.00.4e50.00.00.00.20.00.10.00.10.4e60.20.00.00.00.10.00.00.10.4e70.00.00.10.00.00.00.00.10.1e80.00.00.00.00.10.10.10.00.3Table 9
Tabular representation ofHe2 with choice values.
e1e2e3e4e5e6e7e8Ci2e10.00.30.00.20.00.00.10.00.6e20.30.00.00.00.00.00.00.10.4e30.00.00.00.20.00.00.10.00.3e40.20.00.20.00.10.00.00.00.5e50.00.00.00.10.00.10.00.10.3e60.00.00.00.00.10.00.00.00.1e70.10.00.10.00.00.00.00.00.2e80.00.10.00.00.10.00.00.00.2The decision value isSi=∨i8∧kCik=∨i=180.5∧0.6,0.6∧0.4,0.4∧0.3,0.4∧0.5,0.4∧0.3,0.4∧0.1,0.1∧0.2,0.3∧0.2=0.5 from the choice value Cik of fuzzy graph Hek for k=1,2. Clearly, the dominant object is e1 or e4. 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.


## 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 graphG=Φ,Ψ,M, where Φ,M is a fuzzy soft set over V=e1,e2,e3,e4,e5,\\e6,e7,e8 which describes the membership of the objects based upon the given parameters e1 and e2; Ψ,M is a fuzzy soft set over E=e1e2,e1e4,e2e3,e3e4,e4e5,e5e6,e5e6,e5e8,e6e7,e8e7⊆V×V describing the membership between two objects corresponding to the given parameters e1 and e2. A fuzzy soft graph G=He1,He2 is given in Tables 6 and 7.Table 6
Tabular representation of a fuzzy soft vertex set.
Φe1e2e3e4e5e6e7e8e10.80.70.20.40.30.30.20.2e20.80.50.50.40.40.50.60.5Table 7
Tabular representation of a fuzzy soft edge set.
Ψe1e2e1e6e1e4e1e7e2e3e2e4e2e8e3e4e3e7e4e5e5e6e5e8e7e8e10.30.20.00.00.20.10.00.10.10.20.10.10.1e20.30.00.20.10.00.00.10.20.10.20.10.10.0The fuzzy graphsHe1 and He2 of fuzzy soft graph G=He1,He2 corresponding to the parameters “carbohydrates” and “protein” are shown in Figure 30.Figure 30
Fuzzy soft graphG=He1,He2.The fuzzy graphsHe1 and He2 and the choice values Cik=∑jSij for all i,j,k=1,2 are given in Tables 8 and 9, respectively.Table 8
Tabular representation ofHe1 with choice values.
e1e2e3e4e5e6e7e8Ci1e10.00.30.00.00.00.20.00.00.5e20.30.00.20.10.00.00.00.00.6e30.00.20.00.10.00.00.10.00.4e40.00.10.10.00.20.00.00.00.4e50.00.00.00.20.00.10.00.10.4e60.20.00.00.00.10.00.00.10.4e70.00.00.10.00.00.00.00.10.1e80.00.00.00.00.10.10.10.00.3Table 9
Tabular representation ofHe2 with choice values.
e1e2e3e4e5e6e7e8Ci2e10.00.30.00.20.00.00.10.00.6e20.30.00.00.00.00.00.00.10.4e30.00.00.00.20.00.00.10.00.3e40.20.00.20.00.10.00.00.00.5e50.00.00.00.10.00.10.00.10.3e60.00.00.00.00.10.00.00.00.1e70.10.00.10.00.00.00.00.00.2e80.00.10.00.00.10.00.00.00.2The decision value isSi=∨i8∧kCik=∨i=180.5∧0.6,0.6∧0.4,0.4∧0.3,0.4∧0.5,0.4∧0.3,0.4∧0.1,0.1∧0.2,0.3∧0.2=0.5 from the choice value Cik of fuzzy graph Hek for k=1,2. Clearly, the dominant object is e1 or e4. 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.



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