
==== Front
Heliyon
Heliyon
Heliyon
2405-8440
Elsevier

S2405-8440(24)12027-0
10.1016/j.heliyon.2024.e35996
e35996
Research Article
m-Polar interval-valued fuzzy hypergraphs and its application in decision-making problems
Bera Sanchari maths.sanchari@gmail.com
a
Khalaf Osamah Ibrahim usama81818@nahrainuniv.edu.iq
b
Wong Wing-Keung wong@asia.edu.tw
c
Pal Madhumangal mmpalvu@mail.vidyasagar.ac.in
ad⁎
a Department of Applied Mathematics, Vidyasagar University, Midnapore - 721102, India
b Department of Solar, Al-Nahrain Research Center for Renewable Energy, Al-Nahrain University, Jadriya, Baghdad, Iraq
c Department of Finance, Asia University, Taiwan
d Department of Mathematics and Innovation, Saveetha School of Engineering, Chennai-602105, Tamilnadu, India
⁎ Corresponding author at: Department of Applied Mathematics, Vidyasagar University, Midnapore - 721102, India. mmpalvu@mail.vidyasagar.ac.in
13 8 2024
30 8 2024
13 8 2024
10 16 e359962 3 2024
6 8 2024
7 8 2024
© 2024 The Author(s)
2024
https://creativecommons.org/licenses/by-nc-nd/4.0/ This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).
This study introduces the novel concept of m-polar interval-valued fuzzy hypergraph (m-PIVFHG), an advancement that combines the strengths of fuzzy theory and hypergraph models to improve decision-making processes. m-PIVFHGs allow vertices to have degrees of membership across multiple polarities within sub-interval values of [0,1]. This offers better adaptability and precision than traditional models. This paper systematically explores the theoretical foundations of m-PIVFHGs, detailing their unique characteristics and presenting duality concepts with illustrative examples. It also defines various cut and level types specific to m-PIVFHGs and examines their properties. The practical utility of m-PIVFHGs is demonstrated through a real-world application aimed at optimizing decision-making in a university setting, showcasing significant improvements over existing fuzzy graph methodologies.

Keywords

Fuzzy graph
Fuzzy hypergraph
m-Polar fuzzy graph
Interval-valued fuzzy graph
m-Polar interval-valued fuzzy graph
==== Body
pmc1 Introduction

Mathematical modeling is fundamental in understanding both discrete and continuous physical systems, with hypergraphs (HGs) playing a crucial role in applied sciences and technical fields. HGs extend traditional graphs by representing multi-arity relations, useful in partitioning, clustering, and covering tasks in circuit design, and in simulating complex system architectures. The extension to fuzzy theory, particularly through the introduction of m-polar interval-valued fuzzy (m-PIVF) sets, enhances the adaptability and compatibility of these models.

Despite the extensive applications and benefits of hypergraphs, existing literature reveals significant gaps. While interval-valued fuzzy graphs (IVFGs) and m-polar fuzzy graphs (m-PFGs) have been explored separately, their integration into hypergraphs under a multipolar framework remains underdeveloped. This study addresses these gaps by proposing m-polar interval-valued fuzzy hypergraphs (m-PIVFHGs), a novel approach that amalgamates the properties of m-PFGs and IVFGs, thus offering a robust tool for decision-making in uncertain and complex scenarios.

The primary motivation behind this research is to enhance decision-making frameworks in real-world applications where data often comes from multiple sources and exhibits inherent uncertainty, one example of such situation is available in [40]. By allowing interval values in the range [0,1] to describe vertex membership degrees, m-PIVFHGs provide a more flexible and accurate representation compared to traditional models. This is particularly relevant in fields such as social network analysis, medical research, and operational management, where decision-making processes benefit from multi-polar and fuzzy data interpretations.

FHGs can model complex social networks where relationships are not binary but exhibit varying degrees of influence and connection strength. By incorporating m-polar fuzzy sets (m-PFSs), these models can account for multi-faceted social ties and the uncertainty in social interactions. In healthcare, decision-making often involves uncertain and imprecise information. FHGs can represent the relationships between different medical conditions, treatments, and patient responses, allowing for more nuanced and reliable medical decisions. FHGs can model the intricate relationships in supply chain networks, where multiple suppliers, manufacturers, and distributors interact. Incorporating interval-valued fuzzy data helps in better handling the uncertainties in supply and demand, leading to optimized decision-making. In bioinformatics, FHGs can represent the complex interactions among genes, proteins, and other biological entities. The fuzzy nature of these graphs helps in capturing the inherent uncertainty and variability in biological data. FHGs can be used to model ecological systems where multiple environmental factors interact in complex ways. The fuzzy and multi-polar aspects of these graphs allow for better representation and analysis of environmental data, leading to improved management strategies.

In this paper, we present the theoretical foundation of m-PIVFHGs, define various cut and level types, and explore their properties through illustrative examples. We also demonstrate the practical application of m-PIVFHGs in optimizing decision-making processes within a university setting, highlighting the model's advantages over existing methodologies. By addressing the identified research gaps, this work contributes a novel and comprehensive approach to the field of fuzzy hypergraphs, with significant implications for future research and practical applications.

1.1 Literature review

Finding a dominant individual in a social network, conference, meeting, or group debate is necessary in the real world. A robust method for identifying the most influential members of a network is the FG. A ranking procedure that often includes many selection scopes is needed to choose the best options.

Frequently, there are multiple competing factors, which makes the selection process challenging. The proposed graph demonstrates its suitability for handling hazy and ambiguous information that occurs in reality. The FG concept is one of the most prevalent and widely used tools for real-world problem representations, modeling, and analysis. The graph's vertices (nodes) and edges (arcs) define the items and their relationships. Graphs have long been used to represent items and their connections. Many environmental problems and occurrences involve complexities and ambiguities, making communicating certainty challenging.

The use of fuzzy sets was pioneered by Zadeh [44], which helped to resolve these issues. The fuzzy set focuses on an object's membership level in a specific set. Based on Zadeh's fuzzy connection, Kaufman [26] depicted FGs. The structure of FGs used to derive analogs for various graph theoretical ideas was outlined by Rosenfeld [39]. Mordeson and Nair [34] developed several FG principles. The necessity for a degree of membership was felt since the lack of a single degree for actual membership could not resolve the uncertainty on uncertain subjects. FGs are currently being used extensively in research projects, including link prediction in social networks [28], [29], electricity distribution system [15], Indian banking system [16] and generalized neutrosophic planar graphs [31], covering and pair domination in intuitionistic fuzzy graphs [41], isomorphism of m-polar fuzzy graphs [22], cubic graph [38], completeness and regularity of FS [42], edge coloring of FGs [30], product bipolar FGs [23], etc. The basic definitions, terminologies and applications are available in [36].

IVFG was defined by Hongmei and Lianhua [24], who also investigated its characteristics. Edge regular IFGs were discussed by Karunambigai et al. [25]. Bhutani [18] made significant contributions by introducing fuzzy homomorphism, isomorphism, and co-weak isomorphism in 1989. This subject has undergone numerous generalizations since the idea of FGs. Bipolar fuzzy set theory by Zhang [45] forms the foundation for subsequent developments in 1994. The subsequent question arises when the elements of fuzzy sets are represented as intervals. This was also addressed for IVFGs by Hongmei and Lianhua [24] and by Akram and Dudek [1] in 2009 and 2011, respectively.

In specific real-world problems, data comes from multiple sources, indicating the presence of multi-subject experts, multi-trait, and multi-list information that cannot be conveyed mathematically by fuzzy systems alone. In 2014, Chen et al. [19] proposed the influential concept of m-PFS, as an extension of bipolar fuzzy sets, to effectively handle the properties of algebraic and graphical models. Subsequently, Ghorai and Pal [20], [21] examined several properties of m-PFGs such as isomorphic properties and density. Akram et al. [5] investigated the connectivity indices of m-polar fuzzy network models and applied their findings to a product manufacturing problem, demonstrating the model's practical utility in industrial applications. Mondal et al. [32] addressed road network optimization using m-polar fuzzy graphs, emphasizing the models' adaptability in urban planning and logistics. Muhiuddin et al. [33] explored the integrity of m-polar fuzzy graphs, focusing on network stability and security, and showcasing applications in social networks and communication systems. Combining these two types of graphs, i.e., IVFG and m-PFG, Bera and Pal [9], [10], [11], [12] investigated a new kind of graph called m-PIVFG.

In recent years, significant advancements have been made in the application of fuzzy and hypergraph theories to decision-making processes and complex systems modeling. HGs represent a generalization of graphs [13] that incorporate multiary relations. This extension enables the expansion of graph models to effectively capture the complexities of modeling complex systems. When dealing with systems that involve fuzzy binary and multiary relations among objects, transitioning to FHGs is a more natural approach. FHGs combine the strengths of both fuzzy and graph models, facilitating formal optimization and logical procedures. However, employing FGs and HGs as models for various systems (such as social, economic, and communication networks) can present challenges. Fuzzy independent sets, domination fuzzy sets, and fuzzy chromatic sets serve as invariants for analyzing the structural properties of FGs and FHGs under isomorphism transformations [14].

Sarwar et al. [43] introduced a novel group decision-making approach based on rough soft approximations of graphs and HGs, demonstrating its effectiveness in various applied mathematics and computing scenarios in 2023. Similarly, Akram and Nawaz [3] explored the implementation of single-valued neutrosophic soft HGs on the human nervous system, offering new insights into the application of hypergraph theory in biomedical fields. Further contributions by Akram et al. [4] proposed decision-making methods based on fuzzy soft competition hypergraphs, highlighting their utility in complex and intelligent systems. Additionally, Nawaz et al. [35] developed an algorithm to compute the strength of competing interactions in the Bering Sea using Pythagorean FHGs, emphasizing the robustness of FHG models in environmental and ecological studies.

These studies collectively underscore the versatility and potential of integrating fuzzy sets, soft sets, and HGs in addressing multifaceted problems across diverse disciplines. Our research builds on this foundation by introducing the concept of m-PIVFHG, which aims to further enhance decision-making frameworks by incorporating multi-polarity and interval-valued membership degrees.

1.2 Research gap

Despite the extensive research conducted on HGs and their various applications, significant gaps remain, particularly when integrating m-PFG and IVFGs. Previous studies have explored HGs in the context of IFGs and the multipolar situation in m-PFGs. However, the combination of m-PFG and IVFG has been inadequately addressed, leaving a substantial gap in understanding how these concepts can be effectively applied together. Table 1 highlights the key differences and advantages of our m-PIVFHG model over the existing models such as m-polar fuzzy hypergraphs, transversals of m-polar fuzzy hypergraphs, double dominating energy of m-polar fuzzy graphs, and m-polar fuzzy competition graphs.Table 1 Comparison with related models.

Table 1Feature	m-PFH	Transversals of m-PFH	Double Dominating Energy of m-PFG	Proposed Model	
Handling Uncertainty	Single-valued	Single-valued	Single-valued	Interval-valued	
Number of Poles	m	m	m	m	
Representation of Relationships	Binary	Binary	Binary	Interval-valued	
Application Scope	General	Specific	Specific	General, broader application	
Decision-Making Support	Advanced	Advanced	Advanced	Advanced, interval-valued uncertainty	
Complexity Handling	Moderate	High	High	High, with better uncertainty representation	

The combination of m-PFGs and IVFGs is crucial for addressing complex decision-making and allocation problems in operational research, social networks, communication networks, and other dynamic systems. Specifically, the integration of these two types of graphs can enhance the modeling of systems that require a significant approach to uncertainty and multi-faceted relationships.• In operations research, where decision-making involves multiple criteria and uncertainty, the combination of m-PFG and IVFG can provide a more robust framework. For example, in resource allocation problems within supply chains, the m-PIVFGH model can account for varying degrees of resource availability and demand uncertainty.

• In social network analysis, where relationships and influence levels vary, using m-PIVFGH can improve the accuracy of link prediction and community detection by considering multiple degrees of relationship strength and membership uncertainty.

• For communication networks, the proposed model can help optimize network reliability and performance by handling multiple levels of signal strength and interference more effectively than traditional models.

The novel m-PIVFGH model aims to bridge these gaps by providing a comprehensive approach that leverages the strengths of both m-PFG and IVFG. This model facilitates better decision-making and allocation by accommodating the complex, multi-dimensional nature of real-world data. By doing so, it significantly advances the current methodologies used in various research fields and practical applications

1.3 Motivation

As we know, society's overall well-being is strongly impacted by the existence of universities, which are very significant institutions. We attempt to recognize the most authoritative professor in a university based on their performance because management in each department of the university is crucial. The proposed notions are more flexible than those offered in conventional FG because they allow interval values in [0,1] rather than precise actual values between zero and one to describe the degree of membership of vertices to hyperedges in a HG on IVFG.

HGs are advantageously employed in a variety of fields, including medicine, sociological studies, network modeling, and many others. Since data in real-world situations often comes from multiple factors or numerous sources and cannot be adequately handled by traditional fuzzy systems or other systems, we need m-PFS to manage the data's multiple uncertainties and ambiguities. Furthermore, real-world maximum time figures are more likely to be estimates than precise quantities. Therefore, data should be represented roughly within the interval [0,1]. In this context, IFS is created. The m-PIVF is useful in dealing with the m-tuple properties of an object with certain interval values.

1.4 Main contribution

The primary objectives of this research are as follows:• A novel mathematical model that integrates m-PFS and IVF concepts into hypergraph.

• Demonstrates how m-PIVFHGs can be applied to improve decision-making processes in various real-world scenarios characterized by uncertainty and complex data relationships.

• Provides a detailed exploration of the theoretical properties of m-PIVFHGs, including various cut and level types, duality concepts, and illustrative examples.

• Applies the proposed model to a real-world case study in a university setting, showcasing its effectiveness and improvements over existing fuzzy graph methods.

1.5 Blueprint

The paper is structured as follows: after an introductory section, we go over some primers and critical theories with illustrative instances in Section 2. The detailed analysis of the presentation of all the definitions and intriguing aspects of the new notion of HG in m-PIVFG, which includes m-PIVFHG and dual m-PIVFHG, is presented in Section 3. Following that, a few distinct types of cut levels in m-PIVFHG are described with suitable examples in Section 4. The results for several types of m-PIVFHG are shown in Section 5. A discussion of a real-world case study on m-PIVFHG follows. The current approach and strategy are also considered, and the suggested plan is examined in Section 6. We conclude the paper by providing a summary of future directions and a graph-theoretic conclusion for the entire work in Section 7. The abbreviations used in this study are listed in Appendix A.

2 Preliminaries

In this section, we briefly explain the basic terminologies related to HGs, m-PFGs, IVFGs, and m-PIVFGs utilized throughout this study. All the related notations are provided in Table 2.Table 2 Notations.

Table 2Symbol	Description	
G⁎	Crisp graph	
G	Fuzzy graph	
H = (V,D)	Hypergraph	
H′ = (V′,D′)	Dual hypergraph	
V = {t1,t2,⋯,tr}	Vertex set for G, H	
E	Family of subsets of V, that means edge set for G, H	
D = {F1,F2,⋯,Fs}	An m-PIVF relation on subsets of V	
i = {1,2,⋯,r}	Set of all possible choices for vertices of H	
j = {1,2,⋯,s}	Set of all possible choices for edges of H	
k = {1,2,⋯,m}	Set of all possible choices for poles of G and H	
νAl(t), νAu(t)	Lower and upper membership values for the vertex t	
qi∘ν(t)	Membership value of ith pole for the vertex t	
V′ = {f1,f2,⋯,fs}	Vertex set for H′	
D′ = {T1,T2,⋯,Tr}	Edge set for H′	
O(H),S(H),N(S)	Order and size of H, cardinality of any set S	

FS, an extension of classical set theory, was introduced by Zadeh in 1965 [44]. A FS is defined as a collection of elements with associated degrees of membership. Represented as (S,ν), where S denotes the underlying set and ν:S→[0,1] is the membership function.

Definition 2.1 [27] A hypergraph H is defined as H=(V,D), where:(i) V={t1,t2,...,tr},

(ii) D={F1,F2,...,Fs},

(iii) Each hyperedge Fj is non empty set, indicated as Fj≠∅,

(iv) The union of all hyperedges Fj covers all vertices in V, expressed as ∪jFj=V.

Definition 2.2 [27] The edge Fj is visually represented by a solid line enclosing its vertices when |Fj|=1, or by a cycle containing the elements when |Fj|=2 for all j. In the case where |Fj|=2 for all j, the HG is equivalent to a standard undirected graph. Alternatively, the hypergraph (V,D) can be denoted as (V;F1,F2,…,Fs) to represent the vertex set V along with its corresponding hyperedges F1,F2,…,Fs.

Definition 2.3 [27] In an HG, two edges Fi and Fj are considered adjacent if their intersection is non-empty, denoted as Fi∩Fj≠∅, where i≠j.

Definition 2.4 [27] In an HG, the degree of a vertex t is determined by the number of edges that contain the vertex.

Definition 2.5 [27] In a hypergraph H=(V,D), the incidence matrix (Im) of H is a matrix MH=(aij)r×s. The matrix has s columns representing the edges and r rows representing the vertices. Each element aij in the matrix indicates the membership of a vertex to a hyperedge as follows:aij={1;if xi∈Fj,0;ifxi≠Fj.

Definition 2.6 [27] A hypergraph H={(X;F1,F2,⋯,Fs)} with vertex set X={t1,t2,⋯,tn} can be transformed into a dual hypergraph H′=(F;t1,t2,⋯,tr). In the dual HG, the vertices correspond to the hyperedges F1,F2,⋯,Fs and the edges correspond to the sets {T1,T2,...,Tr}, where each Ti is defined as {Ti=fj|ti∈Fj}. It is important to note that Ti is non-empty and that the union of all Ti equals the set of hyperedges D. The dual hypergraph H′ is the resulting HG obtained through this mapping process.

Definition 2.7 [10] An m-PIVFG of a crisp graph G⁎ is defined as a triplet G=(V,A,B), where V is a non-empty set, A:V→[0,1]m is an m-PFS defined on V, and B:V×V→[0,1]m and ν:V×V→[0,1]m are interval-valued functions. For each k, qk∘νA(t)=[qk∘νAl(t),qk∘νAu(t)] with 0≤νAl(t)≤νAu(t)≤1. Similarly, qk∘νB(ty)=[qk∘νBl(ty),qk∘νBu(ty)] with 0≤νBl(ty)≤νBu(ty)≤1, satisfying, qk∘νBl(ty)≤qk∘min⁡{νAl(t),νAl(y)} and qk∘νBu(ty)≤qk∘min⁡{νAu(t),νAu(y)}. These conditions hold true for all t,y∈V.

The upcoming section provides explanations of the dual HG within the framework of m-PIVFG, accompanied by relevant examples.

3 Hypergraph in m-PIVFG

In this section, we introduce the concept of HGs in m-PIVFG, illustrating it with examples. Additionally, we explore the notion of dual HGs in m-PIVFG through appropriate examples.

Definition 3.1 The concept of an m-PIVFHG on a non-empty set X is defined by H=(V,D), where V={t1,t2,⋯tr} represents a family of m-PIVF subsets on X, and D={F1,F2,⋯,Fs} represents an m-PIVF relation on the m-PIVF subsets of V such that(1) D(Fj)={(ti,<[qk∘νjl(ti),qk∘νju(ti)]>):[qk∘νjl(ti),qk∘νju(ti)]⊂[0,1] and [qk∘νjl(ti),qk∘νju(ti)]>[0,0]}

(2) Fj≠ϕ.

(3) D(Fj)={(ti,<[qk∘νjl(ti),qk∘νju(ti)]> where, qk∘νjl(ti)≤inf⁡{qk∘νjl(t1),qk∘νjl(t2),⋯,qk∘νjl(tn)} and

qk∘νju(ti)≤inf⁡{qk∘νju(t1),qk∘νju(t2),⋯,qk∘νju(tn)} and ∀t1,t2,⋯,tn∈V and qk∘νjl(ti)⩽qk∘νju(ti).

(4) ∪jSuppFj=V, where SuppFj={ti:ti∈V and [qk∘νjl(ti),qk∘νju(ti)]>[0,0]}, for each i.

Example 3.1 Let H=(V,D) be an m-PIVFHG (Fig. 1) where, V={t1,t2,t3,t4,t5,t6} and D={F1,F2,F3,F4} with the membership value given in Table 3.Figure 1 A 3-PIVFHG H.

Figure 1

Table 3 Membership value for Fi of ti.

Table 3ti	F1	F2	F3	F4	
t1	<[0.2,0.6],[0.3,0.5],[0.1,0.4]>	–	–	–	
t2	<[0.1,0.4],[0.2,0.5],[0.5,0.6]>	<[0.1,0.4],[0.2,0.5],[0.5,0.6]>	–	–	
t3	–	–	<[0.0,0.2],[0.2,0.3],[0.4,0.5]>	<[0.0,0.2],[0.2,0.3],[0.4,0.5]>	
t4	<[0.1,0.6],[0.0,0.4],[0.6,0.7]>	<[0.1,0.6],[0.0,0.4],[0.6,0.7]>	–	<[0.1,0.6],[0.0,0.4],[0.6,0.7]>	
t5	–	<[0.2,0.8],[0.7,0.9],[0.1,0.5]>	<[0.2,0.8],[0.7,0.9],[0.1,0.5]>	–	
t6	–	–	<[0.4,0.9],[0.0,0.7],[0.2,0.3]>	–	

From the definition and direct calculation, the hyper-edges are

D(F1)=F{t1,t2,t4}=<[0.1,0.4],[0.0,0.4],[0.1,0.4]>.

D(F2)=F{t2,t4,t5}=<[0.0,0.3],[0.0,0.4],[0.1,0.5]>.

D(F3)=F{t3,t5,t6}=<[0.0,0.2],[0.0,0.1],[0.1,0.3]>.

D(F4)=F{t3,t4}=<[0.0,0.1],[0.0,0.3],[0.4,0.5]>.

Definition 3.2 Any H′=(V′,D′) be dual m-PIVFHG of H where V′={f1,f2,⋯,fs} corresponding to {F1,F2,⋯,Fs} and D′={T1,T2,⋯,Tr} corresponding to {t1,t2,⋯,tr} respectively and Tj={(fi,<[qk∘νjl(fi),qk∘νju(fi)]>)|qk∘νjl(fi)=qk∘νil(Tj) and qk∘νju(fi)=qk∘νiu(Tj)}.

Example 3.2 From Example 3.1, dual H′ (see Fig. 2) of H is H′=(V′,D′) with V′={f1,f2,f3,f4} and D′={T1,T2,T3,T4,T5,T6}. Calculating dual hyperedges, we obtain:Figure 2 A dual 3-PIVFHG H′ of H.

Figure 2

T1={(f1,<[0.2,0.6],[0.3,0.5],[0.1,0.4]>)},

T2={(f1,<[0.1,0.4],[0.2,0.5],[0.5,0.6]>),(f2,<[0.1,0.4],[0.2,0.5],[0.5,0.6]>)},

T3={(f3,<[0.0,0.2],[0.2,0.3],[0.4,0.5]>),(f4,<[0.0,0.2],[0.2,0.3],[0.4,0.5]>)},

T4={(f1,<[0.1,0.6], [0.0,0.4],[0.6,0.7]>),(f2,<[0.1,0.6],[0.0,0.4],[0.6,0.7]>), (f4,<[0.1,0.6],[0.0,0.4],[0.6,0.7]>)},

T5={(f2,<[0.2,0.8],[0.7,0.9],[0.1,0.5]>),(f3,<[0.2,0.8],[0.7,0.9],[0.1,0.5]>)} and T6={(f3,<[0.4,0.9],[0.0,0.7],[0.2,0.3]>)}. Hence, the Im are given in the Table 4.Table 4 Im of fi and Ti.

Table 4Ti	f1	f2	f3	f4	
T1	<[0.2,0.6],[0.3,0.5],[0.1,0.4]>	–	–	–	
T2	<[0.1,0.4],[0.2,0.5],[0.5,0.6]>	<[0.1,0.4],[0.2,0.5],[0.5,0.6]>	–	–	
T3	–	–	<[0.0,0.2],[0.2,0.3],[0.4,0.5]>	<[0.0,0.2],[0.2,0.3],[0.4,0.5]>	
T4	<[0.1,0.6],[0.0,0.4],[0.6,0.7]>	<[0.1,0.6],[0.0,0.4],[0.6,0.7]>	–	<[0.1,0.6],[0.0,0.4],[0.6,0.7]>	
T5	–	<[0.2,0.8],[0.7,0.9],[0.1,0.5]>	<[0.2,0.8],[0.7,0.9],[0.1,0.5]>	–	
T6	–	–	<[0.4,0.9],[0.0,0.7],[0.2,0.3]>	–	

4 Different cuts in m-PIVFHG

In this section, different cuts at various levels in m-PIVFHGs are characterized and illustrated with appropriate examples, along with the formulation of several hypotheses.

Definition 4.1 A crisp valued α-cut at β-level hypergraph Hα,β of an m-PIVFHG H can be obtained by first cutting the m-PIVFHG H at α-level to derive an m-PFHG Hα⁎ and then employing β-cut to Hα⁎ to acquire α-cut β-level Hα,β′ where α∈[0,1] and β∈[0,1].

Let H be an m-PIVFHG, H=(V,D). Let Fj be an hyperedge in H, Fj∈D whereEj={(t1,<[a11l,a11u],[a12l,a12u],⋯,[a1ml,a1mu]>)(t2,<[a21l,a21u],[a22l,a22u],⋯,[a2ml,a2mu]>),

⋯,(tn,<[an1l,an1u],[an2l,an2u],⋯,[anml,anmu]>)}.

First, by applying α-cut to H, the α-cut Fj, α of the hyperedge Fj of H can be obtained whereFj,α={(t1,<p11,p12,⋯,p1m>),(t2,<p21,p22,⋯,p2m>),⋯,(tn,<pn1,pn2,⋯,pnm>)}

with pik∈[0,1] where 1≤i≤n and 1≤k≤m, 1≤j≤s. The values of pik represent the degree to which the membership of ti to Fj is greater than α, where α∈[0,1], 1≤i≤n andpik={1;if α≤aikl,aiku−αaiku−aikl;ifaikl≤α≤aiku,0;ifaiku≤α.

After applying the α-cut operation, it becomes apparent that an m-PIVFHG Hα⁎=(Vα⁎,Dα⁎), where Vα⁎={t1,t2,⋯,tn} and Dα⁎={F1,α,F2,α,⋯,Fs,α} with Fj,α={(t1,<p11,p12,⋯, p1m>),(t2,<p21,p22,⋯,p2m>),⋯,(tn,<pn1,pn2,⋯,pnm>)}. Now applying β-cut we get Fj,α={xj,<q1,q2,⋯,qm>}, qj is either 0 or 1. Now,qj={1;if β≤pik,0;ifβ>pik.

Thus, applying β-cut to Hα⁎ we get α-cut at β-level m-PIVFHG Hα,β⁎=(Vα,β⁎,Dα,β⁎) of H with Vα,β⁎={t1,t2,⋯,tn} and Dα,β⁎={F1,α,β,F2,α,β,⋯,Fs+1,α,β} where Fj,α,β={xi;qk∘νj(xi)≠0foranyk} and Fj,α,β={0;qk∘νj(ti)=0forallk} with k=1,2,⋯,m, j=1,2,⋯s+1.

Example 4.1 From Example 3.1, let H=(V,D) be an m-PIVFHG with V={t1,t2,t3,t4,t5,t6} and D={F1,F2,F3,F4} where

F1={(t1,<[0.2,0.6],[0.3,0.5],[0.1,0.4]>),(t2,<[0.1,0.4],[0.2,0.5],[0.5,0.6]>), (t4,<[0.1,0.6],[0.0,0.4],[0.6,0.7]>)}.

F2={(t2,<[0.1,0.4],[0.2,0.5],[0.5,0.6]>),(t4,<[0.1,0.6],[0.0,0.4],[0.6,0.7]>), (t5,<[0.2,0.8],[0.7,0.9],[0.1,0.5]>)}.

F3={(t3,<[0.0,0.2],[0.2,0.3],[0.4,0.5]>),(t5,<[0.2,0.8],[0.7,0.9],[0.1,0.5]>), (t6,<[0.4,0.9],[0.0,0.7],[0.2,0.3]>)}.

F4={(t3,<[0.0,0.2],[0.2,0.3],[0.4,0.5]>),(t4,<[0.1,0.6],[0.0,0.4],[0.6,0.7]>)}.

Now, find 0.5-cut at 0.4-level on H as shown in Table 5. First, applying 0.5-cut, we get:Table 5 Im of ti and Fi,α.

Table 5ti	F1,0.5	F2,0.5	F3,0.5	F4,0.5	F5,0.5	
t1	<0.25,0,0>	–	–	–	–	
t2	<0,0,1>	<0,0,1>	–	–	–	
t3	–	–	–	–	–	
t4	<0.2,0,1>	<0.2,0,1>	–	<0,0,1>	–	
t5	–	<0.5,1,0>	<0.5,1,0>	–	–	
t6	–	–	<0.8,0.29,0>	–	–	

F1,0.50={(t1,<0.25,0.0,0.0>),(t2,<0.0,0.0,1.0>),(t4,<0.2,0.0,1.0>)}.

F2,0.50={(t2,<0.0,0.0,1.0>),(t4,<0.2,0.0,1.0>),(t5,<0.50,1.0,0.0>)}.

F3={(t5,<0.50,1.0,0.0),(t6,<0.8,0.29,0.0>)}.F4={(t4,<0.2,0.0,1.0>)}. Now applying 0.4-level cut, we obtain Table 6 (Fig. 3), which isTable 6 Im of ti and Fi,α,β.

Table 6ti	F1,0.5,0.4	F2,0.5,0.4	F3,0.5,0.4	F4,0.5,0.4	F5,0.5,0.4	
t1	0	–	–	–	1	
t2	1	1	–	–	–	
t3	–	–	–	–	1	
t4	1	1	–	1	–	
t5	–	1	1	–	–	
t6	–	–	1	–	–	

Figure 3 α-cut β-level 3-PIVFHG H.

Figure 3

F1,0.5,0.4={(t1,<0,0,0>),(t2,<0,0,1>),(t4,<0,0,1>)}={(t1,0),(t2,1),(t4,1)}={t2,t4}.

F2,0.5,0.4={(t2,<0,0,1>),(t4,<0,0,1>),(t5,<1,1,0>)}={(t2,1),(t4,1),(t5,1)}={t2,t4,t5}.

F3={(t5,<1,1,0),(t6,<1,0,0>)}={(t5,1),(t6,1)}={t5,t6}.

F4={(t4,<0,0,1>)}={(t4,1)}={t4}andF5={t1,t3}.

Remark 1 An α-cut β-level dual m-IVFHG is the same as α-cut β-level m-IVFHG. Here, we simply apply α-cut β-level on the dual m-IVFHG H′=(V′,D′) of m-IVFHG H=(V,D).

Example 4.2 From Example 3.2, let H′=(V′,D′) be a dual m-PIVFHG of H=(V,D) with V′={f1,f2,f3,f4} and D′={T1,T2,T3,T4,T5,T6} where

T1={(f1,<[0.2,0.6],[0.3,0.5],0.1,0.4]>)}.

T2={(f1,<[0.1,0.4],[0.2,0.5],[0.5,0.6]>),(f2,<[0.1,0.4],[0.2,0.5],[0.5,0.6]>)}.

T3={(f3,<[0.0,0.2],[0.2,0.3],[0.4,0.5]>),(f4,<[0.0,0.2],[0.2,0.3],[0.4,0.5]>)}.

T4={(f1,<[0.1,0.6],[0.0,0.4],[0.6,0.7]>),(f2,<[0.1,0.6],[0.0,0.4],[0.6,0.7]>), (f4,<[0.1,0.6],[0.0,0.4],[0.6,0.7]>)}.

T5={(f2,<[0.2,0.8],[0.7,0.9],[0.1,0.5]>),(f3,<[0.2,0.8],[0.7,0.9],[0.1,0.5]>)}.

T6={(f3,<[0.4,0.9],[0.0,0.7],[0.2,0.3]>)}.

First, applying the 0.5-cut, the obtain outcomes are provided in Table 7 (see Fig. 4). Next, applying the 0.4-level cut on H0.5′, the derived values are given in Table 8.Table 7 Im of fi and Ti,α.

Table 7fi	T1,0.5	T2,0.5	T3,0.5	T4,0.5	T5,0.5	T6,0.5	
f1	<0.25,0,0>	<0,0,1>	–	<0.2,0,1>	–	–	
f2	–	<0,0,1>	–	<0.2,0,1>	<0.5,1,0>	–	
f3	–	–	–	–	<0.5,1,0>	<0.8,0.285,0>	
f4	–	–	–	<0.2,0,1>	–	–	

Figure 4 α-cut β-level dual 3-PIVFHG H′.

Figure 4

Table 8 Im of fi and Ti,α,β.

Table 8fi	T1,0.5,0.4	T2,0.5,0.4	T3,0.5,0.4	T4,0.5,0.4	T5,0.5,0.4	T6,0.5,0.4	
f1	–	1	–	1	–	–	
f2	–	1	–	1	1	–	
f3	–	–	–	–	1	1	
f4	–	–	–	1	–	–	

Definition 4.2 An interval-valued [α1,α2]-cut at β-level hypergraph H[α1,α2],β′ of an IVFHG, H can be obtained by first cutting the IVFHG, H at [α1,α2]-cut to derive an FHG H[α1,α2]⁎ and then applying β-cut to H[α1,α2]⁎ to get [α1,α2]-cut at β-level H[α1,α2],β′ where [α1,α2]∈[0,1] and β∈[0,1].

Let Fj be a hyperedge of an IVFHG H=(V,D), Fj∈D where Fj={(t1,<[a1l,a1u]),(t2,<[a2l,a2u]),⋯,(tn,<[anl,anu]>)}. First, by applying α-cut to H, the α-cut Fj, α of the hyperedge Fj of H can be obtained where Fj,[α1,α2]={(t1,k1),(t2,k2),⋯,(tn,kn)} with ki∈[0,1] where 1≤i≤n and 1≤j≤m. Now, ki=pi1+pi2+⋯+pinn, where taking α=α1,pi1={1;if α≤ail,aiu−αaiu−ail;ifail≤α≤aiu,0;ifaiu≤α.

Next, by taking α=α1+0.01, we get, pi2, and we repeat this process until α=α2, yielding pin. It is obvious that after applying α-cut operation, we get an FHG Hα⁎=(V[α1,α2]⁎,D[α1,α2]⁎), where V[α1,α2]⁎={t1,t2,⋯,tn} and D[α1,α2]⁎={F1,[α1,α2],F2,[α1,α2],⋯,Fs,[α1,α2]} with Fj,[α1,α2]={(t1,k1),(t2,k2),⋯,(tn,kn)}. Now applying β-cut, we get Fj,[α1,α2]={tj,qj}, where qj is either 0 or 1.qj={1;if β≤ki,0;ifβ>ki.

Thus, by applying β-cut to H[α1,α2]⁎, we obtain [α1,α2]−c at β-level IVFHG H[α1,α2],β⁎=(V[α1,α2],β⁎,Dα,β⁎) of H with V[α1,α2],β⁎={t1,t2,⋯,tn} and D[α1,α2],β⁎={F1,[α1,α2],β,F2,[α1,α2],β,⋯,

Fs+1,[α1,α2],β}, where Fj,[α1,α2],β={(ti,qi);qi≠0foranyi} and Fj,[α1,α2],β={0;qi=0∀i} with i=1,2,⋯,n, j=1,2,⋯s+1.

Example 4.3 Let H=(V,D) be an IVFHG with V={t1,t2,t3,t4,t5,t6} and D={F1,F2}. The hyperedges are defined as follows:F1={(t1,[0.3,0.6]),(t3,[0.4,0.8]),(t5,[0.1,0.3])},

F2={(t2,[0.2,0.5]),(t4,[0.2,0.4]),(t6,[0.6,0.7])}.

Therefore, the degrees of the hyperedges are D(F1)=[0.3,0.6] and D(F2)=[0.2,0.4]. Fig. 5 illustrates the structure of the IVFHG H, showing the vertices and the interval-valued memberships for each hyperedge.Figure 5 An IVFHG H.

Figure 5

Now, we find 0.5-cut at 0.4-level on H.(i). First, we find k1 for t1=[0.3,0.6]:• When α1=0.4: P11=0.6−0.40.6−0.3=0.67

• Then, α1+0.01=0.41: P12=0.6−0.410.6−0.3=0.63

• α1+0.01=0.42: P13=0.6−0.420.6−0.3=0.60

• Continue this process until α1=0.5: P14=0.57, P15=0.53, P16=0.5, P17=0.47, P18=0.43, P19=0.4, P110=0.37, P111=0.33.

• Hence, k1=0.67+0.63+0.6+0.57+0.53+0.5+0.47+0.43+0.4+0.37+0.3311=0.53.

(ii). Similarly, find k2 for t2=[0.2,0.5]:• When α1=0.4: P21=0.33.

• Then, α1+0.01=0.41: P22=0.30.

• Continue this process until α1=0.5: P23=0.27, P24=0.23, P25=0.20, P26=0.17, P27=0.13, P28=0.10, P29=0.07, P210=0.03, P211=0.

• Hence, k2=0.33+0.3+0.27+0.23+0.2+0.17+0.13+0.1+0.07+0.03+011=0.17.

(iii). Similarly, find k3 for t3=[0.4,0.8]:• When α1=0.4: P31=1.

• Then, α1+0.01=0.41: P22=0.975, P33=0.95, P34=0.925, P35=0.9, P36=0.875, P37=0.85, P38=0.825, P39=0.8, P310=0.775, P311=75.

• Hence, k3=1+0.975+0.95+0.925+0.9+0.875+0.85+0.825++0.8+0.775+0.7511=0.875.

(iv). For t4=[0.2,0.4]: Since, α1=0.4≥0.4=x4u, so k4=0.

(v). For t5=[0.1,0.3]: Since, α1=0.4>0.3=x5u, so k5=0.

(vi). For t6=[0.6,0.7]: Since, α1=0.4<0.6=x6l, so k6=1.

Therefore, after applying the [α1,α2]=[0.4.0.5]-cut, we get F1,[0.4,0.5]={(t1,0.53),(t3,0.875),(t5,0)} and F2={(t2,0.17),(t4,0),(t6,1)}, as shown in Table 9.Table 9 Im of ti and Fi,[α1,α2].

Table 9ti	F1,[0.4,0.5]	F2,[0.4,0.5]	
t1	0.53	–	
t2	–	0.17	
t3	0.875	–	
t4	–	0	
t5	0	–	
t6	–	1	

Now, we apply a 0.6-level cut on H[0.4,0.5]′, resulting in H[0.4,0.5],0.6, which is detailed in Table 10. Fig. 6 provides a pictorial representation. The sets are defined as F1,[0.4,0.5],0.6={t1,t3}, F2,[0.4,0.5],0.6={t6} and F3,[0.4,0.5],0.6={t2,t4,t5}.Table 10 Im of ti and Fi,[α1,α2],β.

Table 10ti	F1,[0.4,0.5],0.6	F2,[0.4,0.5],0.6	F3,[0.4,0.5],0.6	
t1	1	–	–	
t2	–	–	1	
t3	1	–	–	
t4	–	–	1	
t5	–	–	1	
t6	–	1	–	

Figure 6 [α1,α2]−c at β-level of an IVFHG H.

Figure 6

Definition 4.3 An interval-valued [α1,α2]-cut at β-level hypergraph H[α1,α2],β′ of an m-PIVFHG H can be obtained by first cutting the m-PIVFHG H at [α1,α2]−c to derive an m-PFHG H[α1,α2]⁎ and then employing a β-cut to H[α1,α2]⁎ to acquire [α1,α2]-cut β-level H[α1,α2],β′ where [α1,α2]∈[0,1] and β∈[0,1].

Let Fj be an hyperedge in H, where Fj={(t1,<[a1l,a1u]),(t2,<[a2l,a2u]),⋯,(tn,<[anl,anu]>)}. First, by applying the α-cut to H, the α-cut Fj, α of the hyperedge Fj of H can be obtained where Fj,[α1,α2]={(t1,k1),(t2,k2),⋯,(tn,kn)} with ki∈[0,1] where 1≤i≤n and 1≤j≤m. The value of qik is given by:qik=pik1+pik2+⋯+piknn.

Taking α1=α, pik1={1;if α≤aikl,aiku−αaiku−aikl;ifaikl≤α≤aiku,0;ifaiku≤α.

Next, taking α=α1+0.01, we get pik2, similarly repeat this process until α=α2, we get pikn.

After applying α-cut operation, we get an FHG Hα⁎=(V[α1,α2]⁎,D[α1,α2]⁎), where V[α1,α2]⁎={t1,t2,⋯,tn} and D[α1,α2]⁎={F1,[α1,α2],F2,[α1,α2],⋯,Fs,[α1,α2]} with Fj,[α1,α2]={(t1,<q1k>),(t2,<q2k>),⋯,(tj,<qjk>),⋯,(tn,<qnk>)}.

Now, applying β-cut, we get Fj,[α1,α2]={tj,qj}, where qj is either 0 or 1.

The value qik is given by:qik={1;if β≤qik0,ifqik≤β

Thus, applying β-cut to H[α1,α2]⁎, we get [α1,α2]-cut at β-level IVFHG H[α1,α2],β⁎=(V[α1,α2],β⁎,Dα,β⁎) of H with V[α1,α2],β⁎={t1,t2,⋯,tn} and D[α1,α2],β⁎={F1,[α1,α2],β,F2,[α1,α2],β,⋯, Fs+1,[α1,α2],β} where Fj,[α1,α2],β={(ti,qi);qi≠0foranyi} and Fj,[α1,α2],β={0;qi=0foralli} with i=1,2,⋯,n, j=1,2,⋯s+1 and 1≤k≤m.

Example 4.4 To find the [0.45,0.50]-cut at 0.4-level for Example 3.2, using the definitions provided, we need to find F1,[0.45,0.50], F2,[0.45,0.50], F3,[0.45,0.50] and F4,[0.45,0.50] (see Table 11). Here are the steps involved:Table 11 Im of Fi,[α1,α2] and ti.

Table 11ti	F1,[0.45,0.5]	F2,[0.45,0.5]	F3,[0.45,0.5]	F4,[0.45,0.5]	
t1	<0.31,0.13,0>	–	–	–	
t2	<0,0.08,1>	<0,0.08,1>	–	–	
t3	–	–	<0,0,0.25>	<0,0,0.25>	
t4	<0.25,0,1>	<0.25,0,1>	–	<0.25,0,1>	
t5	–	<0.46,1,0.06>	<0.46,1,0.06>	–	
t6	–	–	<0.85,0.32,0>	–	

Step 1: Calculate qik for each vertex ti 1. For t1: α1=0.45, α2=0.5: P111=0.375, P112=0.35, P113=0.325, P114=0.3, P115=0.275, P116=0.25. Therefore, q11=0.375+0.35+0.325+0.3+0.275+0.256=0.3125. Similarly, q12=0.125,q13=0. Thus for t1, we get (t1,<0.3125,0.125,0>).

2. For t2: q21=0,q22=0.083,q23=1. Thus, for t2: (t2,<0,0.083,1>).

3. For t3: q31=0,q32=0,q33=0.25. Thus, for t3: (t3,<0,0,0.25>).

4. For t4: q41=0.25,q42=0,q43=1. Thus, for t4: (t4,<0.25,0,1>).

5. For t5: q51=0.46,q52=1,q53=0.06. Thus, for t5: (t5,<0.46,1,0.06>).

6. For t6: q61=0.85,q62=0.32,q63=0. Thus, for t6: (t6,<0.85,0.32,0>).

Step 2: ConstructFj,[0.45,0.5]sets: Using the calculated qik values, we obtain the following sets:• F1,[0.45,0.5]={(t1,<0.3125,0.125,0>),(t2,<0,0.083,1>),(t4,<0.25,0,1>)}

• F2,[0.45,0.5]={(t2,<0,0.083,1>),(t4,<0.25,0,1>),(t5,<0.46,1,0.06>)}

• F3,[0.45,0.5]={(t3,<0,0,0.25>),(t5,<0.46,1,0.06>),(t6,<0.85,0.32,0>)}

• F4,[0.45,0.5]={(t3,<0,0,0.25>),(t4,<0.25,0,1>)}

We now find β=0.4-level on each Ei,[α1,α2] for i=1,2,3,4. These values are detailed in Table 12 and illustrated in Fig. 7.Table 12 Im of Fi,[α1,α2],β and ti.

Table 12ti	F1,[0.45,0.5],0.4	F2,[0.44,0.5],0.4	F3,[0.45,0.5],0.4	F4,[0.45,0.5],0.4	
t1	–	–	–	–	
t2	1	1	–	–	
t3	–	–	–	–	
t4	1	1	–	1	
t5	–	1	1	–	
t6	–	–	1	–	

Figure 7 A 3-PIVFHG H.

Figure 7

Definition 4.4 An m-pole [α1,α2,⋯,αm]-cut at β-level H[α1,α2,⋯,αm],β′ of an m-PIVFHG H can be obtained by first applying the m-PIVFHG H at [α1,α2,⋯,αm]-level to derive an m-PFHG H[α1,α2,⋯,αm]⁎ and then performing the β-cut on H[α1,α2,⋯,αm]⁎ to obtain [α1,α2,⋯,αm]-cut β-level H[α1,α2,⋯,αm],β′ where αk∈[0,1] and β∈[0,1] for k=1,2,⋯,m.

Let H be an m-PIVFHG, H=(V,D). Let Fj be a hyperedge in H, Fj∈D where Fj={(ti,<[ai1l,ai1u]),[ai2l,ai2u],⋯,[ainl,ainu]>)}. Applying [α1,α2,⋯,αm]−c to H, we obtain the [α1,α2,⋯,αm]-cut Fj,[α1,α2,⋯,αm] of hyperedge Fj of H, given by Fj,[α1,α2,⋯,αm]={(t1,q1k),(t2,q2k),⋯,(tn,qnk)}, where qik∈[0,1] and 1≤i≤n, 1≤k≤m.

Specifically, qi1={1;if α1≤ai1lai1u−α1ai1u−ai1l;ifai1l≤α1≤ai1u0;ifai1u≤α1, and similarly for qi1,⋯,qim.

After employing the [α1,α2,⋯,αm]-cut operation, we obtain an m-PFHG H[α1,α2,⋯,αm]⁎=(V[α1,α2,⋯,αm]⁎,D[α1,α2,⋯,αm]⁎), where V[α1,α2,⋯,αm]⁎={t1,t2,⋯,tn} and D[α1,α2,⋯,αm]⁎={F1,[α1,α2,⋯,αm], F2,[α1,α2,⋯,αm],⋯,Fs,[α1,α2,⋯,αm]} with Fj,[α1,α2,⋯,αm]={(t1,<q11,q12,⋯,q1m>),(t2,<q21,q22,⋯, q2m>),⋯,(tn,<qn1,qn2,⋯,qnm>)}.

Subsequently, applying the β-cut yields Fj,[α1,α2,⋯,αm],β={tj,qj}, where qj is either 0 or 1, determined byqj={1;if β≤qik0.ifqik≤β.

Thus, by applying the β-cut to H[α1,α2,⋯,αm]⁎, we obtain the [α1,α2,⋯,αm]-cut at β-level m-PIVFHG, H[α1,α2,⋯,αm],β⁎=(V[α1,α2,⋯,αm],β⁎,D[α1,α2,⋯,αm],β⁎) of H with V[α1,α2,⋯,αm],β⁎={t1,t2,⋯,tn} and D[α1,α2,⋯,αm],β⁎={F1,[α1,α2,⋯,αm],β,F2,[α1,α2,⋯,αm],β,⋯,Fs+1,[α1,α2,⋯,αm],β} where Fj,[α1,α2,⋯,αm],β={(ti,qi);qi≠0foranyi} and Fj,[α1,α2,⋯,αm],β={0;qi=0∀i}.

Example 4.5 From the Example 3.2, let H=(V,D) be a 3-PIVFHG with V={t1,t2,t3,t4} and D={F1,F2,F3,F4}. The hyperedges are given as:

F1={(t1,<[0.2,0.6],[0.3,0.5],[0.1,0.4]>),(t2,<[0.1,0.4],[0.2,0.5],[0.5,0.6]>), (t4,<[0.1,0.6],[0.0,0.4],[0.6,0.7]>)},

F2={(t2,<[0.1,0.4],[0.2,0.5],[0.5,0.6]>),(t5,<[0.2,0.8],[0.7,0.9],[0.1,0.5]>)},

F3={(t3,<[0.0,0.2],[0.2,0.3],[0.4,0.5]>),(t5,<[0.2,0.8], [0.7,0.9],[0.1,0.5]>), (t6,<[0.4,0.9],[0.0,0.7],[0.2,0.3]>)},

F4={((t3,<[0.0,0.2],[0.2,0.3],[0.4,0.5]>),(t4,<[0.1,0.6],[0.0,0.4],[0.6,0.7]>)}.

Now, find the <0.3,0.5,0.6>-cut at 0.4-level on H.

First, applying <0.3,0.5,0.6>-cut, we get:• For vertex t1: <0.6−0.30.4,0.5−0.50.5−0.3,0.0>=<0.75,0.0,0.0>.

• For vertex t2: <0.4−0.30.4−0.1,0.5−0.50.5−0.2,0.6−0.60.6−0.5>=<0.33,0.0,0.0>.

• For vertex t3: <0.0,0.0,0.0>.

• For vertex t4: <0.6−0.30.6−0.1,0.0,0.7−0.60.7−0.6>=<0.6,0.0,1.0>.

• For vertex t5: <0.8−0.30.8−0.2,1.0,0.0>=<0.833,1.0,0.0>.

• For vertex t6: <1.0,0.7−0.50.7,0.0>=<1.0,0.286,0.0>.

Thus, the cut hyperedges are:

F1,<0.3,0.5,0.6>={(t1,<0.75,0,0>),(t2,<0.33,0.0,0.0>),(t4,<0.6,0.0,1.0>)}.

F2,<0.3,0.5,0.6>={(t2,<0.33,0.0,0.0>),(t4,<0.6,0.0,1.0>),(t5,<0.833,1.0,0.0>)}.

F3,<0.3,0.5,0.6>={(t3,<0.0,0.0,0.0>),(t5,<0.833,1.0,0.0>),(t6,<1.0,0.286,0.0>)}.

F4,<0.3,0.5,0.6>={(t3,<0.0,0.0,0.0>),(t4,<0.6,0.0,1.0>)}.

Thus, we get H<0.3,0.5,0.6>⁎=(V<0.3,0.5,0.6>⁎,F<0.3,0.5,0.6>⁎), where F<0.3,0.5,0.6>⁎={F1,<0.3,0.5,0.6>,F2,<0.3,0.5,0.6>,F3,<0.3,0.5,0.6>,F4,<0.3,0.5,0.6>} (refer to Table 13).Table 13 Im of ti and Fi,<0.3,0.5,0.6>.

Table 13ti	F1,<0.3,0.5,0.6>	F2,<0.3,0.5,0.6>	F3,<0.3,0.5,0.6>	F4,<0.3,0.5,0.6>	
t1	<0.75,0,0>	–	–	–	
t2	<0.33,0,0>	<0.33,0,0>	–	–	
t3	–	–	<0,0,0>	<0,0,0>	
t4	<0.6,0,1>	<0.6,0,1>	–	<0.6,0,1>	
t5	–	<0.833,1,0>	<0.833,1,0>	–	
t6	–	–	<1,0.286,0>	–	

We now apply the 0.4-cut on H<0.3,0.5,0.6>⁎:

F1,<0.3,0.5,0.6>,0.4={(t1,<1,0,0>),(t2,<0,0,0>),(t4,<1,0,1>)}={t1,t4}

F2,<0.3,0.5,0.6>,0.4={(t2,<0,0,0>),(t4,<1,0,1>),(t5,<1,1,0>)}={t4,t5},

F3,<0.3,0.5,0.6>,0.4={(t3,<0,0,0>),(t5,<1,1,0>),(t6,<1,0,0>)}={t5,t6},

F4,<0.3,0.5,0.6>,0.4={(t3,<0,0,0>),(t4,<1,0,1>)}={t4} and F5,<0.3,0.5,0.6>,0.4={t2,t3}.

Finally, we obtain the <0.3,0.5,0.6>-cut at 0.4-level 3-PIVFHG

H<0.3,0.5,0.6>,0.4⁎=(V<0.3,0.5,0.6>,0.4⁎,F<0.3,0.5,0.6>,0.4⁎), where F<0.3,0.5,0.6>,0.4⁎={F1,<0.3,0.5,0.6>,0.4={t1,t4},F2,<0.3,0.5,0.6>,0.4={t4,t5},F3,<0.3,0.5,0.6>,0.4={t5,t6},F4,<0.3,0.5,0.6>,0.4}={t4},F5,<0.3,0.5,0.6>,0.4={t2,t3} (see Table 14, Fig. 8).Table 14 Im of ti and Fi,<0.3,0.5,0.6>,0.4.

Table 14ti	F1,<0.3,0.5,0.6>,0.4	F2,<0.3,0.5,0.6>,0.4	F3,<0.3,0.5,0.6>,0.4	F4,<0.3,0.5,0.6>,0.4	F5,<0.3,0.5,0.6>,0.4	
t1	<1,0,0>	—	—	—	—	
t2	<0,0,0>	<0,0,0>	—	—	1	
t3	—	—	<0,0,0>	<0,0,0>	1	
t4	<1,0,1>	<1,0,1>	—	<1,0,1>	—	
t5	—	<1,1,0>	<1,1,0>	—	—	
t6	—	—	<1,0,0>	—	—	

Figure 8 <0.3,0.5,0.6>-cut at 0.4-level on a 3-PIVFG H.

Figure 8

5 Different properties

In this topic, numerous properties such as homomorphism, weak-isomorphism, co-weak-isomorphism, and isomorphism are explained with various suitable instances.

Definition 5.1 A homomorphism between two m-PIVFHGs H and H′ is a mapping ϕ:V→V′ with V={t1,t2,⋯,tr} and V′={t1′,t2′,⋯,tr′} such that(i) For each hyperedge Fj(t), the following inequalities hold:

Fj(t)≤Fj′(ϕ(t)), that is, qk∘νl(Fj(t))≤qk∘νl(Fj′(ϕ(t))), qk∘νu(Fj(t))≤qk∘νu(Fj′(ϕ(t))).

(ii) For each set of vertices {t1,t2,⋯,tr} in V:

D(Fj{t1,t2,⋯,tr})≤D(Fj′{ϕ(t1),ϕ(t2),⋯,ϕ(tr)}), ∀t∈V.

Example 5.1 Let H be a 3-PIVFHG defined as H=(V,D) depicted in Fig. 9a, where V={t1,t2,t3,t4,t5,t6,t7} and D={F1,F2,F3} with the membership values given in the Table 15. The hyperedges {F1,F2,F3} are defined as follows:Figure 9 Illustration of the homomorphism process for a 3-PIVFHG: (9a) Initial subgraph H and (9b) Mapped subgraph H′.

Figure 9

Table 15 Im of ti and Fi.

Table 15ti	F1	F2	F3	
t1	<[0.1,0.2],[0.3,0.6],[0.1,0.3]>	—	—	
t2	—	<[0.2,0.4],[0.2,0.3],[0.4,0.6]>	—	
t3	—	—	<[0.2,0.6],[0.1,0.3],[0.3,0.4]>	
t4	<[0.2,0.3],[0.4,0.5],[0.2,0.6]>	—	—	
t5	—	<[0.3,0.5],[0.4,0.5],[0.5,0.6]>	—	
t6	—	—	<[0.2,0.4],[0.6,0.8],[0.4,0.6]>	
t7	<[0.4,0.5],[0.6,0.7],[0.1,0.4]>	—	—	

F1={(t1,<[0.1,0.2],[0.3,0.6],[0.1,0.3]>),(t4,<[0.2,0.3],[0.4,0.5],[0.2,0.6]>), (t7,<[0.4,0.5],[0.6,0.7],[0.1,0.4]>)}.

F2={(t2,<[0.2,0.4],[0.2,0.3],[0.4,0.6]>),(t5,<[0.3,0.5],[0.4,0.5],[0.5,0.6]>)}.

F3={(t3,<[0.2,0.6],[0.1,0.3],[0.3,0.4]>),(t6,<[0.2,0.4],[0.6,0.8],[0.4,0.6]>)}.

Similarly, let H′ be another a 3-PIVFHG depicted Fig. 9b, defined as H′=(V′,D′) where V′={t1′,t2′,t3′,t4′,t5′,t6′,t7′} and D′={F1′,F2′,F3′} with the membership values given in Table 16. The hyperedges {F1′,F2′,F3′} are defined as follows:Table 16 Im of ti′ and Fi′.

Table 16ti′	F1′	F2′	F3′	
t1′	<[0.2,0.3],[0.3,0.7],[0.3,0.5]>	—	—	
t2′	—	<[0.2,0.4],[0.2,0.3],[0.4,0.6]>	—	
t3′	—	—	<[0.3,0.6],[0.1,0.4],[0.3,0.5]>	
t4′	—	—	<[0.2,0.4],[0.6,0.8],[0.4,0.6]>	
t5′	—	<[0.3,0.5],[0.4,0.5],[0.5,0.6]>	—	
t6′	<[0.2,0.3],[0.4,0.5],[0.5,0.6]>	—	—	
t7′	<[0.4,0.5],[0.6,0.7],[0.1,0.4]>	—	—	

F1′={(t1′,<[0.2,0.3],[0.3,0.7],[0.3,0.5]>),(t7′,<[0.4,0.5],[0.6,0.7],[0.1,0.4]>)}.

F2′={(t2′,<[0.2,0.4],[0.2,0.3],[0.4,0.6]>),(t5′,<[0.3,0.5],[0.4,0.5],[0.5,0.6]>)}.

F3′={(t3′,<[0.3,0.6],[0.1,0.4],[0.3,0.5]>),(t4′,<[0.2,0.4],[0.6,0.8],[0.4,0.6]>), (t6′,<[0.2,0.3],[0.4,0.5],[0.5,0.6]>)}.

Consider a mapping ϕ:V→V′ such that:qk∘νl(ϕ(t1))=qk∘νl(t1′),qk∘νu(ϕ(t1))=qk∘νu(t1′),

qk∘νl(ϕ(t2))=qk∘νl(t2′),qk∘νu(ϕ(t2))=qk∘νu(t2′),

qk∘νl(ϕ(t3))=qk∘νl(t3′),qk∘νu(ϕ(t3))=qk∘νu(t3′),

qk∘νl(ϕ(t4))=qk∘νl(t6′),qk∘νu(ϕ(t4))=qk∘νu(t6′),

qk∘νl(ϕ(t5))=qk∘νl(t5′),qk∘νu(ϕ(t5))=qk∘νu(t5′),

qk∘νl(ϕ(t6))=qk∘νl(t4′),qk∘νu(ϕ(t6))=qk∘νu(t4′),

qk∘νl(ϕ(t7))=qk∘νl(t7′),qk∘νu(ϕ(t7))=qk∘νu(t7′).

Thus, we have:

F1(t1)=<[0.1,0.2],[0.3,0.6],[0.1,0.3]>≤<[0.2,0.3],[0.3,0.7],[0.3,0.5]>=F1′(ϕ(t1))=F1′(t1′),

F1(t4)=<[0.2,0.3],[0.4,0.5],[0.2,0.6]>=F1′(ϕ(t4))=F1′(t6′),

F1(t7)=<[0.4,0.5],[0.6,0.7],[0.1,0.4]>=F1′(ϕ(t7))=F1′(t7′),

F2(t2)=<[0.2,0.4],[0.2,0.3],[0.4,0.6]>=F2′(ϕ(t2))=F2′(t2′),

F2(t5)=<[0.3,0.5],[0.4,0.5],[0.5,0.6]>=F2′(ϕ(t5))=F2′(t5′),

F3(t3)=<[0.2,0.6],[0.1,0.3],[0.3,0.4]>≤<[0.3,0.6],[0.1,0.4],[0.3,0.5]>=F3′(ϕ(t3))=F3′(t3′)),

F3(t6)=<[0.2,0.4],[0.6,0.8],[0.4,0.6]>≤<[0.2,0.4],[0.6,0.8],[0.4,0.6]>=F3′(ϕ(t3))=F3′(t4′).

Therefore, Fj(t)≤Fj′(ϕ(t)).

Furthermore, consider the degrees:

D(F1)=<[0.1,0.2],[0.3,0.6],[0.1,0.3]>,

D(F1′)=<[0.2,0.3],[0.3,0.5],[0.1,0.4]>⇒D(F1)≤D(F1′),

D(F2)=<[0.2,0.4],[0.2,0.3],[0.1,0.4]>=D(F1′),

D(F3)=<[0.2,0.4],[0.1,0.3],[0.1,0.4]>D(F3′)=<[0.2,0.4],[0.1,0.4],[0.1,0.4]>⇒D(F3)≤D(F3′).

Thus, D(Fj{t1,t2,⋯t7})≤D(Fj′{ϕ(t1),ϕ(t2),⋯ϕ(t7)}). Hence, there exists a homomorphism ϕ:H→H′.

Definition 5.2 A weak isomorphism of between two m-PIVFHGs H=(V,D) and H′=(V′,D′) is a mapping ϕ:V→V′ such that(i) Fj(t)=Fj′(ϕ(t)), that is, qk∘νl(Fj(t))=qk∘νl(Fj′(ϕ(t))), qk∘νu(Fj(t))=qk∘νu(Fj′(ϕ(t))), for all t∈V, k, and j.

(ii) D(Fj{t1,t2,⋯,tr})≤D(Fj′{ϕ(t1),ϕ(t2),⋯,ϕ(tr)}), ∀t∈V and for each k and j.

This condition establishes that it is a special type of homomorphism where Fj(t)=Fj′(ϕ(t)).

Example 5.2 Consider a 2-PIVFHG H (Fig. 10a) with V={t1,t2,t3,t4,t5,t6} and D={F1,F2,F3,F4} with the membership values given in Table 17. The edges {F1,F2,F3,F4} are defined as:Figure 10 Demonstration of the weak-isomorphism process for a 2-PIVFHG: (10a) Initial subgraph H and (10b) Mapped subgraph H′.

Figure 10

Table 17 Im of ti and Fi.

Table 17ti	F1	F2	F3	F4	
t1	<[0.1,0.2],[0.3,0.6]>	<[0.1,0.2],[0.3,0.6]>	—	—	
t2	<[0.1,0.2],[0.3,0.6]>	—	—	—	
t3	<[0.1,0.2],[0.3,0.6]>	<[0.1,0.2],[0.3,0.6]>	—	—	
t4	—	—	<[0.2,0.3],[0.2,0.6]>	<[0.2,0.3],[0.2,0.6]>	
t5	—	—	—	<[0.3,0.5],[0.5,0.6]>	
t6	—	<[0.2,0.4],[0.4,0.6]>	— <[0.4,0.5],[0.1,0.4]>		

F1={(t1,<[0.3,0.4][0.5,0.6]>),(t2,<[0.1,0.4],[0.2,0.6]>),(t3,<[0.2,0.5],[0.2,0.3]>)},

F2={(t1,<[0.3,0.4],[0.5,0.6]>),(t6,<[0.3,0.4],[0.4,0.6]>)},

F3={(t3,<[0.2,0.5],[0.2,0.3]>),(t4,<[0.1,0.3],[0.3,0.5]>)},

F4={(t4,<[0.1,0.3],[0.3,0.5]>),(t5,<[0.3,0.5],[0.2,0.7]>),(t6,<[0.3,0.4],[0.4,0.6]>)}.

Let H′ be another 2-PIVFHG (Fig. 10b) defined as H′=(V′,D′) where, V′={t1′,t2′,t3′,t4′,t5′,t6′} and D′={F1′,F2′,F4′} with the membership values given in Table 18. The edges {F1′,F2′,F3′,F3′} are defined as:Table 18 Im of ti′ and Fi′.

Table 18ti′	F1′	F2′	F3′	F4′	
t1′	—	—	<[0.1,0.3],[0.3,0.5]>	<[0.1,0.3],[0.3,0.5]>	
t2′	<[0.1,0.4],[0.2,0.6]>	—	—	—	
t3′	<[0.2,0.5],[0.2,0.3]>	—	<[0.2,0.5],[0.2,0.3]>	—	
t4′	<[0.3,0.4],[0.5,0.6]>	<[0.3,0.4],[0.5,0.6]>	—	—	
t5′	<[0.3,0.4],[0.4,0.6]>	—	—	<[0.3,0.4],[0.4,0.6]>	
t6′	—	—	— <[0.3,0.5],[0.2,0.7]>		

F1′={(t2′,<[0.1,0.4],[0.2,0.6]>),(t3′,<[0.2,0.5],[0.2,0.3]>),(t4′,<[0.3,0.4],[0.5,0.6]>)},

F2′={(t4′,<[0.3,0.4],[0.5,0.6]>),(t5′,<[0.3,0.4],[0.4,0.6]>)},

F3′={(t1′,<[0.1,0.3],[0.3,0.5]>),(t3′,<[0.2,0.5],[0.2,0.3]>)},

F4′={(t1′,<[0.1,0.3],[0.3,0.5]>),(t5′,<[0.3,0.4],[0.4,0.6]>),(t6′,<[0.3,0.5],[0.2,0.7]>)}.

Consider a mapping ϕ:V→V′, defined by:qk∘νl(ϕ(t1))=qk∘νl(t4′),qk∘νu(ϕ(t1))=qk∘νu(t4′),

qk∘νl(ϕ(t2))=qk∘νl(t2′),qk∘νu(ϕ(t2))=qk∘νu(t2′),

qk∘νl(ϕ(t3))=qk∘νl(t3′),qk∘νu(ϕ(t3))=qk∘νu(t3′),

qk∘νl(ϕ(t4))=qk∘νl(t1′),qk∘νu(ϕ(t4))=qk∘νu(t1′),

qk∘νl(ϕ(t5))=qk∘νl(t6′),qk∘νu(ϕ(t5))=qk∘νu(t6′)

qk∘νl(ϕ(t6))=qk∘νl(t5′),qk∘νu(ϕ(t6))=qk∘νu(t5′).

Then, it follows that:

F1(t1)=<[0.3,0.4],[0.5,0.6]>=F1′(ϕ(t1))=F1′(t4′),

F1(t2)=<[0.1,0.4],[0.2,0.6]>=F1′(ϕ(t2))=F1′(t2′),

F1(t3)=<[0.2,0.5],[0.2,0.3]>=F1′(ϕ(t3))=F1′(t3′),

F2(t1)=<[0.3,0.4],[0.5,0.6]>=F2′(ϕ(t1))=F2′(t4′),

F2(t6)=<[0.3,0.4],[0.4,0.6]>=F2′(ϕ(t6))=F2′(t5′),

F3(t3)=<[0.2,0.5],[0.2,0.3]>=F3′(ϕ(t3))=F3′(t3′),

F4(t4)=<[0.1,0.3],[0.3,0.5]>=F4′(ϕ(t4))=F4′(t1′),

F4(t5)=<[0.3,0.5],[0.2,0.7]>=F4′(ϕ(t5))=F4′(t6′),

F4(t6)=<[0.3,0.4],[0.4,0.6]>=F4′(ϕ(t6))=F4′(t5′).

Thus, we have shown that Fj(t)=Fj′(ϕ(t)). Moreover,

D(F1)=<[0.1,0.2],[0.2,0.3]>, D(F1′)=<[0.1,0.4],[0.2,0.3]>⇒D(F1)≤D(F1′),

D(F2)=<[0.1,0.2],[0.2,0.3]>,D(F2′)=<[0.3,0.4],[0.4,0.6]>⇒D(F1)≤D(F1′),

D(F3)=<[0.1,0.2],[0.1,0.2]>, D(F3′)=<[0.1,0.3],[0.2,0.3],[0.1,0.4]>⇒D(F3)≤D(F3′),

D(F4)=<[0.1,0.2],[0.1,0.2]>, D(F4′)=<[0.0,0.2],[0.2,0.4]>⇒D(F4)≤D(F4′).

Hence, D(Fj{t1,t2,⋯t6})≤D(Fj′{ϕ(t1),ϕ(t2),⋯ϕ(t6)}). This implies that there exists a weak isomorphism ϕ:H→H′.

Definition 5.3 Given any two m-PIVFHGs H=(V,D) and H′=(V′,D′) where V={t1,t2,⋯,tr} and V′={t1′,t2′,⋯,tr′}, a co-weak isomorphism of between them is a mapping ϕ:V→V′ such that:(i) Fj(t)≤Fj′(ϕ(t)), that is, qk∘νl(Fj(t))≤qk∘νl(Fj′(ϕ(t))), qk∘νu(Fj(t))≤qk∘νu(Fj′(ϕ(t))).

(ii) D(Fj{t1,t2,⋯,tr})=D(Ej′{ϕ(t1),ϕ(t2),⋯,ϕ(tr)}), ∀t∈V and for each k, j.

This definition describes a special type of homomorphism with the additional condition:

D(Fj{t1,t2,⋯,tr})=D(Fj′{ϕ(t1),ϕ(t2),⋯,ϕ(tr)}).

Example 5.3 Let H=(V,D) be a 2-PIVFHG (Fig. 11a), where V={t1,t2,t3,t4,t5} and D={F1,F2,F3,F4} with the membership values given in Table 19. The edges {F1,F2,F3,F4} are designated as:Figure 11 Illustration of the co-weak-isomorphism process for a 2-PIVFHG: (11a) Initial subgraph H and (11b) Mapped subgraph H′.

Figure 11

Table 19 Im of ti and Fi.

Table 19ti	F1	F2	F3	F4	
t1	<[0.2,0.4],[0.3,0.6]>	—	<[0.2,0.4],[0.3,0.6]>	—	
t2	<[0.3,0.5],[0.4,0.7]>	<[0.3,0.5],[0.4,0.7]>	—	—	
t3	—	<[0.5,0.6][0.5,0.7]>	—	<[0.5,0.6][0.5,0.7]>	
t4	—	<[0.5,0.7][0.6,0.8]>	—	—	
t5	—	—	<[0.1,0.9][0.2,0.3]>	<[0.1,0.9][0.2,0.3]>	

F1={(t1,<[0.3,0.4][0.5,0.6]>),(t2,<[0.1,0.4],[0.2,0.6]>),(t3,<[0.2,0.5],[0.2,0.3]>)},

F2={(t1,<[0.3,0.4],[0.5,0.6]>),(t6,<[0.3,0.4],[0.4,0.6]>)},

F3={(t3,<[0.2,0.5],[0.2,0.3]>),(t4,<[0.1,0.3],[0.3,0.5]>)},

F4={(t4,<[0.1,0.3],[0.3,0.5]>),(t5,<[0.3,0.5],[0.2,0.7]>),(t6,<[0.3,0.4],[0.4,0.6]>)}.

Let H′ be another 2-PIVFHG (Fig. 11b) defined as H′=(V′,D′) where, V′={t1′,t2′,t3′,t4′,t5′} and D′={F1′,F2′,F3′,F4′} with the membership values given in Table 20. The edges {F1′,F2′,F3′,F4′} are addressed as:Table 20 Im of ti′ and Fi′.

Table 20ti′	F1′	F2′	F3′	F4′	
t1′	—	<[0.4,0.6],[0.5,0.8]>	—	<[0.4,0.6],[0.5,0.8]>	
t2′	—	—	<[0.2,0.9],[0.4,0.5]>	<[0.2,0.9],[0.4,0.5]>	
t3′	<[0.2,0.5],[0.3,0.6]>	—	<[0.2,0.5],[0.3,0.6]>	—	
t4′	—	<[0.6,0.7],[0.6,0.8]>	—	—	
t5′	<[0.3,0.5],[0.4,0.8]>	<[0.3,0.5],[0.4,0.8]>	—	—	

F1′={(t2′,<[0.1,0.4],[0.2,0.6]>),(t3′,<[0.2,0.5],[0.2,0.3]>),(t4′,<[0.3,0.4],[0.5,0.6]>)},

F2′={(t4′,<[0.3,0.4],[0.5,0.6]>),(t5′,<[0.3,0.4],[0.4,0.6]>)},

F3′={(t1′,<[0.1,0.3],[0.3,0.5]>),(t3′,<[0.2,0.5],[0.2,0.3]>)},

F4′={(t1′,<[0.1,0.3],[0.3,0.5]>),(t5′,<[0.3,0.4],[0.4,0.6]>),(t6′,<[0.3,0.5],[0.2,0.7]>)}.

Consider a mapping ϕ:V→V′ defined by:qk∘νl(ϕ(t1))=qk∘νl(t3′),qk∘νu(ϕ(t1))=qk∘νu(t3′),

qk∘νl(ϕ(t2))=qk∘νl(t5′),qk∘νu(ϕ(t2))=qk∘νu(t5′),

qk∘νl(ϕ(t3))=qk∘νl(t1′),qk∘νu(ϕ(t3))=qk∘νu(t1′),

qk∘νl(ϕ(t4))=qk∘νl(t4′),qk∘νu(ϕ(t4))=qk∘νu(t4′),

qk∘νl(ϕ(t5))=qk∘νl(t2′),qk∘νu(ϕ(t5))=qk∘νu(t2′).

Here, D(F1)=<[0.1,0.3],[0.2,0.5]>=D(F1′), D(F2)=<[0.2,0.5],[0.3,0.5]>=D(F2′), D(F3)=<[0.2,0.4],[0.2,0.3]>=D(F3′), D(F4)=<[0.1,0.6],[0.2,0.3]>=D(F4′). Hence, there exists a co-weak isomorphism ϕ:H→H′.

Definition 5.4 An isomorphism of among two m-PIVFHGs H=(V,D) and H′=(V′,D′) is a mapping ϕ:V→V′ such that:(i) Fj(t)=Fj′(ϕ(t)), i.e., qk∘νl(Fj(t))=qk∘νl(Fj′(ϕ(t))), qk∘νu(Fj(t))=qk∘νu(Fj′(ϕ(t))).

(ii) D(Fj{t1,t2,⋯,tr})=D(Fj′{ϕ(t1),ϕ(t2),⋯,ϕ(tr)}), ∀t∈V and for each k and j.

Example 5.4 Let H be a 2-PIVFHG (Fig. 12a) with V={t1,t2,t3,t4,t5,t6} and D={F1,F2,F3,F4,F5} with the membership values given in Table 21. The edges {F1,F2,F3,F4,F5} are as follows:Figure 12 Visualization of the isomorphism process for a 2-PIVFHG: (12a) Initial subgraph H and (12b) Mapped subgraph H′.

Figure 12

Table 21 Im of ti and Fi.

Table 21ti	F1	F2	F3	F4	
t1	<[0.2,0.4],[0.3,0.6]>	—	<[0.2,0.4],[0.3,0.6]>	—	
t2	<[0.3,0.5],[0.4,0.7]>	<[0.3,0.5],[0.4,0.7]>	—	—	
t3	—	<[0.5,0.6][0.5,0.7]>	—	<[0.5,0.6][0.5,0.7]>	
t4	—	<[0.5,0.7][0.6,0.8]>	—	—	
t5	—	—	<[0.1,0.9][0.2,0.3]>	<[0.1,0.9][0.2,0.3]>	

F1={(t1,<[0.2,0.3][0.4,0.9],[0.5,0.7]>),(t2,<[0.1,0.4],[0.3,0.5],[0.4,0.7]>), (t3,<[0.2,0.4],[0.1,0.3],[0.4,0.6]>)},

F2={(t4,<[0.2,0.5],[0.5,0.6],[0.5,0.6]>),(t5,<[0.4,0.5],[0.5,0.6],[0.3,0.6]>), (t6,<[0.2,0.4],[0.4,0.6],[0.4,0.7]>)},

F3={(t1,<[0.2,0.3],[0.4,0.9],[0.5,0.7]>),(t4,<[0.2,0.5],[0.5,0.6],[0.5,0.6]>)},

F4={(t3,<[0.2,0.4],[0.1,0.3],[0.4,0.6]>)},

F5={(t2,<[0.1,0.4],[0.3,0.5],[0.4,0.7]>),(t5,<[0.2,0.5],[0.5,0.6],[0.3,0.6>)}.

Let H′ be another a 2-PIVFHG (Fig. 12b) defined as H′=(V′,D′), where V′={t1′,t2′,t3′,t4′,t5′,t6′} and D′={F1′,F2′,F3′,F4′} with the membership values provided in Table 22. The edges {F1′,F2′,F3′,F4′} are as below:Table 22 Im of ti′ and Fi′.

Table 22ti′	F1′	F2′	F3′	F4′	
t1′	—	<[0.4,0.6],[0.5,0.8]>	—	<[0.4,0.6],[0.5,0.8]>	
t2′	—	—	<[0.2,0.9],[0.4,0.5]>	<[0.2,0.9],[0.4,0.5]>	
t3′	<[0.2,0.5],[0.3,0.6]>	—	<[0.2,0.5],[0.3,0.6]>	—	
t4′	—	<[0.6,0.7],[0.6,0.8]>	—	—	
t5′	<[0.3,0.5],[0.4,0.8]>	<[0.3,0.5],[0.4,0.8]>	—	—	

F1′={(t2′,<[0.1,0.4],[0.2,0.6]>),(t3′,<[0.2,0.5],[0.2,0.3]>),(t4′,<[0.3,0.4],[0.5,0.6]>)},

F2′={(t4′,<[0.3,0.4],[0.5,0.6]>),(t5′,<[0.3,0.4],[0.4,0.6]>)},

F3′={(t1′,<[0.1,0.3],[0.3,0.5]>),(t3′,<[0.2,0.5],[0.2,0.3]>)},

F4′={(t1′,<[0.1,0.3],[0.3,0.5]>),(t5′,<[0.3,0.4],[0.4,0.6]>),(t6′,<[0.3,0.5],[0.2,0.7]>)}.

Consider a mapping ϕ:V→V′ by:qk∘νl(ϕ(t1))=qk∘νl(t2′),qk∘νu(ϕ(t1))=qk∘νu(t2′),

qk∘νl(ϕ(t2))=qk∘νl(t4′),qk∘νu(ϕ(t2))=qk∘νu(t4′),

qk∘νl(ϕ(t3))=qk∘νl(t5′),qk∘νu(ϕ(t3))=qk∘νu(t5′),

qk∘νl(ϕ(t4))=qk∘νl(t3′),qk∘νu(ϕ(t4))=qk∘νu(t3′),

qk∘νl(ϕ(t5))=qk∘νl(t1′),qk∘νu(ϕ(t5))=qk∘νu(t1′),

qk∘νl(ϕ(t6))=qk∘νl(t6′),qk∘νu(ϕ(t6))=qk∘νu(t6′).

Thus, we have:

F1(t1)=<[0.2,0.3],[0.4,0.9],[0.5,0.7]>=F1′(ϕ(t1))=F1′(t2′),

F1(t2)=<[0.1,0.4],[0.3,0.5],[0.4,0.7]>=F1′(ϕ(t2))=F1′(t4′),

F1(t3)=<[0.2,0.4],[0.1,0.3],[0.4,0.6]>=F1′(ϕ(t3))=F1′(t5′),

F2(t4)=<[0.2,0.5],[0.5,0.6],[0.5,0.6]>=F2′(ϕ(t4))=F2′(t3′),

F2(t5)=<[0.4,0.5],[0.5,0.6],[0.3,0.6]>=F2′(ϕ(t5))=E2′(t1′),

F2(t6)=<[0.2,0.4],[0.4,0.6],[0.4,0.7]>=F2′(ϕ(t6))=F2′(t6′),

F3(t1)=<[0.2,0.3],[0.4,0.9],[0.5,0.7]>=F3′(ϕ(t1))=F3′(t2′),

F3(t4)=<[0.2,0.3],[0.5,0.6],[0.5,0.6]>=F3′(ϕ(t4))=F3′(t3′),

F4(t3)=<[0.2,0.4],[0.1,0.3],[0.4,0.6]>=F4′(ϕ(t3))=F4′(t5′),

F5(t2)=<[0.1,0.4],[0.3,0.5],[0.4,0.7]>=F5′(ϕ(t2))=F5′(t4′),

F5(t4)=<[0.2,0.5],[0.5,0.6],[0.3,0.6]>=F5′(ϕ(t4))=F5′(t3′).

Furthermore, consider the degrees:

D(F1)=D(F1′)=<[0.1,0.3],[0.1,0.3],[0.4,0.6]>,

D(F2)=D(F2′)=<[0.2,0.4],[0.4,0.6],[0.4,0.6]>,

D(F3)=D(F3′)=<[0.2,0.3],[0.4,0.6],[0.5,0.6]>,

D(F4)=D(F4′)=<[0.2,0.4],[0.1,0.3],[0.4,0.6]>,

D(F5)=D(F5′)=<[0.1,0.4],[0.3,0.5],[0.3,0.6]>.

Hence, there exists an isomorphism ϕ:H→H′.

Theorem 5.1 Isomorphism between any two m-PIVFHGs is an equivalence relation.

Proof Assume three m-PIVFHGs be H=(V,D),H′=(V′,D′)andH″=(V″,D″).i. Reflexive: Let ϕ:V→V be a bijective mapping satisfying ϕ(t)=t, ∀t∈V. Then ϕ satisfies:Ej(t)={(t1,<[a11l,a11u],[a12l,a12u]⋯,[a1ml,a1mu]>),(t2,<[a21l,a21u],[a22l,a22u]⋯,[a2ml,a2mu]>),

⋯,(tn,<[an1l,an1u],[an2l,an2u]⋯,[anml,anmu]>)}

={(ϕ(t1),<[a11l,a11u],[a12l,a12u]⋯,[a1ml,a1mu]>),(ϕ(t2),<[a21l,a21u],[a22l,a22u]⋯,[a2ml,a2mu]>),

⋯,(ϕ(tn),<[an1l,an1u],[an2l,an2u]⋯,[anml,anmu]>)}=Ej′(ϕ(t)),∀t∈V,

andD(Fj)=D(Fj{t1,t2,⋯,tr})={F1,F2,⋯,Fs}={F1′,F2′,⋯,Fs′}=D(Fj′{t1,t2,⋯,tr}).

Hence, ϕ is an isomorphism to itself. Thus, it is a reflexive relation.

ii. Symmetric: Let ϕ:V→V′ be an isomorphism of H and H′ then ϕ is a bijective map satisfying:(5.1) ϕ(t)=t′,∀t∈V.

Then, ϕ is a bijective map satisfying:Fj(t)=Fj′(ϕ(t))D(Fj)=D(Fj{t1,t2,⋯,tr})=D(Fj′{ϕ(t1),ϕ(t2),⋯,ϕ(tr)})∀t∈V.

Since, ϕ is bijective, then by Eq. (5.1), ϕ−1(t′)=t, ∀t′∈V′, and Fj(ϕ−1(t′))=Fj(t)=Fj′(ϕ(t))=Fj′(t′), D(Fj{(ϕ−1(t1′)),(ϕ−1(t2′)),⋯,(ϕ−1(tr′))})=D(Fj′{t1′,t2′,⋯,tr′}).

Hence, we get a 1-1, onto map ϕ−:V′→V. This condition establishes an isomorphism from H′ to H.

iii. Transitive: Let ϕ:V→V′ and ψ:V′→V″ be two isomorphisms of H onto H′ and H′ onto H″, respectively. Then, ψ∘ϕ is 1-1, onto map from V→V″, where ψ∘ϕ(t)=ψ(ϕ(t)), that means ψ∘ϕ(qi∘νl(t))=ψ(ϕ(qi∘νl(t))) and ψ∘ϕ(qi∘νu(t))=ψ(ϕ(qi∘νu(t))) ∀t∈V. As, ϕ:V→V′ is an isomorphism,(5.2) ϕ(t)=t′Fj(t)=Fj′(ϕ(t))D(Fj)=D(Fj{t1,t2,⋯,tr})=D(Fj′{ϕ(t1),ϕ(t2),⋯,ϕ(tr)}).

Hence,(5.3) Fj(t)=Fj′(t′)D(Fj)=D(Fj{t1,t2,⋯,tr})=D(Fj′{t1′,t2′,⋯,tr′}).

Also, as ψ:V′→V″ is an isomorphism,(5.4) ψ(t′)=t″Fj′(t′)=Fj″(ψ(t′))

and(5.5) D(Fj′{t1′,t2′,⋯,tr′})=D(Fj′{ψ(t1″),ψ(t2″),⋯,ψ(tr″)})

Now from Eqs. (5.2) and (5.4) and using ϕ(x)=x′, we get: Fj(x)={(t1,<[a11l,a11u],[a12l,a12u]⋯,[a1ml,a1mu]>),(t2,<[a21l,a21u],[a22l,a22u]⋯,[a2ml,a2mu]>),⋯,(tn,<[an1l,an1u],[an2l,an2u]⋯,[anml,anmu]>)}={(ϕ(t1),<[a11l,a11u],[a12l,a12u]⋯,[a1ml,a1mu]>),(ϕ(t2),<[a21l,a21u],[a22l,a22u]⋯,[a2ml,a2mu]>),⋯,(ϕ(tn),<[an1l,an1u],[an2l,an2u]⋯,[anml,anmu]>)}=Fj′(ϕ(t))=Fj′(t′)=Fj″(ψ(t′))=Fj″(ψ(ϕ(t))), and from Eqs. (5.3) and (5.5),

D(Fj)=D(Fj{t1,t2,⋯,tr})=D(Fj′{t1′,t2′,⋯,tr′})=D(Fj″{ψ(t1′),ψ(t2′),⋯,ψ(tr′)})=D(Fj″{ψ(ϕ(t1)),ψ(ϕ(t2)),⋯,ψ(ϕ(tr))}).

Hence, ψ∘ϕ forms an isomorphism between H and H″.

Thus, isomorphism satisfies the properties of an equivalence relation. □

Theorem 5.2 Weak isomorphism between any two m-PIVFHGs is a partial order relation.

Proof Let H=(V,D),H′=(V′,D′)andH″=(V″,D″) be three m-PIVFHGs.i. Reflexive: Let the identity mapping ϕ:V→V satisfying ϕ(t)=t, ∀t∈V. Then ϕ is a bijective mapping satisfying:

Fj(t)={(t1,<[a11l,a11u],[a12l,a12u]⋯,[a1ml,a1mu]>),(t2,<[a21l,a21u],[a22l,a22u]⋯,[a2ml,a2mu]>),⋯,(tn,<[an1l,an1u],[an2l,an2u]⋯,[anml,anmu]>)}={(ϕ(t1),<[a11l,a11u],[a12l,a12u]⋯,[a1ml,a1mu]>),(ϕ(t2),<[a21l,a21u],[a22l,a22u]⋯,[a2ml,a2mu]>),⋯,(ϕ(tn),<[an1l,an1u],[an2l,an2u]⋯,[anml,anmu]>)}=Fj′(ϕ(t)), ∀t∈V,

and

D(Fj)=D(Fj{t1,t2,⋯,tr})={F1,F2,⋯,Fs}≤{F1′,F2′,⋯,Fs′}=D(Fj′{t1,t2,⋯,tr}). Hence, ϕ is a weak isomorphism to itself. Therefore, H is weakly isomorphic to itself.

ii. Anti-symmetric: Let ϕ:V→V′ and ψ:V′→V be weak isomorphisms between H to H′ and H′ to H, respectively. Then ϕ is a bijective map ϕ(t)=t′ satisfying:(5.6) Fj(t)=Fj′(ϕ(t))D(Fj)=D(Fj{t1,t2,⋯,tr})≤D(Fj′{ϕ(t1),ϕ(t2),⋯,ϕ(tr)})∀t∈V.

and ψ is a bijective map satisfying ψ(t′)=t(5.7) Fj(t′)=Fj′(ψ(t′))D(Fj)=D(Fj{t1′,t2′,⋯,tr′})≤D(Fj′{ψ(t1′),ψ(t2′),⋯,ψ(tr′)})∀t′∈V′.

Eqs. (5.6) and (5.7) hold true only when both H and H′ contain an equal number of edges, each having the same weight. Hence, H and H′ are identical.

iii. Transitive: Let ϕ:V→V′ be a weak-isomorphism from H to H′, and ψ:V′→V″ be a weak-isomorphism from H′ to H″. Then, ψ∘ϕ is 1-1, onto map from V→V″, where ψ∘ϕ(t)=ψ(ϕ(t)). This means ψ∘ϕ(qi∘νl(t))=ψ(ϕ(qi∘νl(t))) and ψ∘ϕ(qi∘νu(t))=ψ(ϕ(qi∘νu(t))) ∀t∈V. As, ϕ:V→V′ is a weak-isomorphism,(5.8) ϕ(t)=t′Fj(t)=Fj′(ϕ(t))

and(5.9) D(Fj)=D(Fj{t1,t2,⋯,tr})≤D(Fj′{ϕ(t1′),ϕ(t2′),⋯,ϕ(tr′)}).

Similarly, since ψ:V′→V″ is weak-isomorphism,(5.10) ψ(t′)=t″Fj′(t′)=Fj″(ψ(t′))

and(5.11) D(Fj′)=D(Fj′{t1,t2,⋯,tr})≤D(Fj′′{ψ(t1′),ψ(t2′),⋯,ψ(tr′)}).

Then, from Eqs. (5.8) and (5.10), we obtain:

Fj(t)=Fj′(ϕ(t))=Fj′(t′)=Fj″(ψ(t′))=Fj″(ψ(ϕ(t)))

and from Eqs. (5.9) and (5.11):

D(Fj)=D(Fj{t1,t2,⋯,tr})≤D(Fj′{ϕ(t1′),ϕ(t2′),⋯,ϕ(tr′)})=D(Fj′{t1,t2,⋯,tr})≤D(Fj′′{ψ(t1′),ψ(t2′),⋯,ψ(tr′)})=D(Fj′′{ψ(ϕ(t1)),ψ(ϕ(t2)),⋯,ψ(ϕ(tr))}).

From above, we conclude that ψ∘ϕ represents a weak-isomorphism. As a result, weak-isomorphism establishes a partial order relation. □

Definition 5.5 Let H=(V,D) be any m-PIVFHG, where V=(t1,t2,⋯,tr) and D={F1,F2,⋯,Fs}. Then order of H is given by:O(H)=∑t∈V1+∑qk∘νl(t)+∑qk∘νu(t)2,

the size of H is designated by:S(H)=∑Fj⊂V1+∑qk∘νDl(Fj)+∑qk∘νDu(Fj)2,

the cardinality of S is defined as:N(S)=∑t∈S1+∑νu(t)−∑νl(t)2,

and the degree of t is as follows:deg(t)=([p1∘degνl(t),p1∘degνu(t)],[p2∘degνl(t),p2∘degνu(t)],⋯,[pm∘degνl(t),pm∘degνu(t)]),

where degνl(t)=∑t∈Fj⊆Vqk∘νDl(Fj), degνu(t)=∑t∈Fj⊆Vqk∘νDu(Fj).

Theorem 5.3 The order and size of any two isomorphic m-PIVFHGs are identical.

Proof Let H1=(V1,D1) and H2=(V2,D2) be any two isomorphic m-PIVFHGs. Suppose V1={t11,t12,⋯,t1r}, V2={t21,t22,⋯,t2r}, D1={F11,F12,⋯,F1s} and D2={F21,F22,⋯,F2s}. Let ϕ:V1→V2 be an isomorphism from H1 to H2. Then, F1j=F2j(ϕ(t)), which means qk∘ν1jl(t)=qk∘ν2jl(ϕ(t)) and qk∘ν1ju(t)=qk∘ν2ju(ϕ(t)), ∀t∈V, j=1,2,⋯,r. D1({t1,t2,⋯,ts})=D2({ϕ(t1),ϕ(t2),⋯,ϕ(ts)}).

Then, we have:O(H1)=∑t1∈V11+qk∘ν1jl(t)+qk∘ν1ju(t)2=∑t1∈V11+qk∘ν1jl(ϕ(t))+qk∘ν1ju(ϕ(t))2=∑t2∈V21+qk∘ν2jl(t)+qk∘ν2ju(t)2=O(H2)

andS(H1)=∑F1j∈V11+qk∘νDl(F1j)+qk∘νDu(F1j)2=∑F1j∈V11+qk∘νDl(ϕ(F1j))+qk∘νDu(ϕ(F1j))2

=∑F2j∈V21+qk∘νDl(F1j)+qk∘νDu(F1j)2=S(H2)

Thus, the proof is complete. □

Theorem 5.4 The degree of the vertices in any two isomorphic m-PIVFHGs remains unchanged.

Proof Let H1=(V1,D1) and H2=(V2,D2) be any two isomorphic m-PIVFHGs, where V1={t11,t12,⋯,t1r}, V2={t21,t22,⋯,t2r}, D1={F11,F12,⋯,F1s} and D2={F21,F22,⋯,F2s}. Using the definition of isomorphism, let ϕ:V1→V2 be an isomorphism from H1 to H2. Then the degree of any vertex t1j∈V1 is given by:deg(t1j)=([p1∘degνl(t1j),p1∘degνu(t1j)],[p2∘degνl(t1j),p2∘degνu(t1j)],⋯,[pm∘degνl(t1j),pm∘degνu(t1j)])=([p1∘degνl(ϕ(t1j)),p1∘degνu(ϕ(t1j))],[p2∘degνl(ϕ(t1j)),p2∘degνu(ϕ(t1j))],⋯,[pm∘degνl(ϕ(t1j)),pm∘degνu(ϕ(t1j))])=([p1∘degνl(t2j),p1∘degνu(t2j)],[p2∘degνl(t2j),p2∘degνu(t2j)],⋯,[pm∘degνl(t2j),pm∘degνu(t2j)])=deg(t2j).

Hence, the degree of the vertices in any two isomorphic m-PIVFHGs remains unchanged. □

6 Case study

Graph theory has proven highly beneficial for addressing combinatorial issues in computer science and communication networks. To better model the complex frameworks encountered in operations research and networking, HGs are introduced to extend the capabilities of traditional graphs. In certain scenarios, the information provided is ambiguous, containing details about both the presence and absence of doubt.

Social network analysis and the study of human nature have long been interrelated. These systems are refined by assigning one or more links to groups of individuals, with relationships derived from successful interactions, management characteristics, or various other sources. Traditional network models described by basic graphs are insufficient for capturing supra-dyadic connections between nodes. Instances where such connections naturally occur include co-citation networks, email networks, co-authorship networks, and web link networks.

IFGs can effectively describe these complex relationships and the inherent ambiguity in the information. Like hypergraphs and fuzzy graphs, IFGs can be utilized to define the existence and non-existence of ambiguity in a more comprehensive manner.

Making decisions is a critical component of modern living. Social networks illustrate the intricate and profound interconnections between various societies. Individuals in society are connected to many different areas, making them part of multiple communities, including workplaces, corporations, educational institutions, healthcare facilities, and universities. We analyze a decision-making problem while keeping in mind these real-world scientific findings.

Consider a scenario where we aim to select the best professor from all the departments in a university based on specific educational criteria and the overall performance of the departments. The task involves categorizing professors based on their field of interest across different departments. Professors are evaluated based on the following criteria:1. Knowledge of the subject

2. Teaching capability

3. Guidance or experience

In this illustration, we consider the professors of a certain university from all departments, and we omit the original names of the professors. We consider 15 professors ‘A’, ‘B’, ‘C’, ‘D’, ‘E’, ‘F’, ‘G’, ‘I’, ‘J’, ‘K’, ‘L’, ‘M’, ‘N’, ‘O’ and ‘P’ from various departments such as ‘Physics’ D(F1), ‘Chemistry’ D(F2), ‘Mathematics’ D(F3), ‘Geography’ D(F4), ‘Micro-biology’ D(F5), ‘Botany’ D(F6), ‘Zoology’ D(F7) and ‘Physiology’ D(F8). Membership values of all vertices are given in Table 23 and each hyperedge is in Table 24.Table 23 Membership value of all vertices t.

Table 23titj	q1	q2	q3	
A	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
B	[0.5,0.6]	[0.3,0.5]	[0.3,0.5]	
C	[0.4,0.5]	[0.4,0.6]	[0.3,0.4]	
D	[0.3,0.4]	[0.5,0.6]	[0.2,0.5]	
E	[0.3,0.4]	[0.5,0.6]	[0.4,0.6]	
F	[0.5,0.6]	[0.5,0.6]	[0.4,0.6]	
G	[0.7,0.8]	[0.6,0.8]	[0.7,0.9]	
I	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	

	titj	q1	q2	q3	
J	[0.5,0.8]	[0.7,0.8]	[0.6,0.8]	
K	[0.4,0.6]	[0.5,0.6]	[0.4,0.5]	
L	[0.3,0.5]	[0.2,0.4]	[0.5,0.6]	
M	[0.4,0.6]	[0.3,0.6]	[0.4,0.5]	
N	[0.5,0.6]	[0.2,0.4]	[0.3,0.7]	
O	[0.4,0.7]	[0.6,0.8]	[0.7,0.8]	
P	[0.3,0.6]	[0.5,0.6]	[0.4,0.5]	

	

Table 24 Membership value of all hyperedges D(Fi).

Table 24D(Fi)	q1	q2	q3	
D(F1)	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
D(F2)	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
D(F3)	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
D(F4)	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
D(F5)	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
D(F6)	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
D(F7)	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
D(F8)	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	

We can present an m-PIVFHG (Fig. 13) in which professors are represented by vertices, and membership values depend on three criteria. Each hyperedge comprises professors who share the same field of interest. For instance, ‘A’, ‘B’ and ‘G’ share a common interest in the subject of mechanics. The membership value of each hyperedge reflects the degree of commonality among these criteria. This m-PIVFHG model can be used to select professors with the best teaching skills. It can also be beneficial for ranking professors considering the aforementioned conditions.Figure 13 A 3-PIVFHG H.

Figure 13

At first, we find the adjacent level (given in Table 25) of two professors Lty=<[qi∘νl(ty),qi∘νu(ty)]>, where qi∘νl(ty)=min⁡{qi∘νl(t),qi∘νl(y)} and qi∘νu(ty)=min⁡{qi∘νu(t),qi∘νu(y)}, for each i=1,2,3. After finding the adjacent level, we determine the score Sty (Table 26) of these two professors using the formulaSty=1+∑i=13qi∘νl(Lty)+∑i=13qi∘νu(Lty)2.

Table 25 Adjacent level of two professors Lxy.

Table 25xixj	q1	q2	q3	
LAB	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
LAG	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
LBG	[0.5,0.6]	[0.3,0.5]	[0.3,0.5]	
LCG	[0.3,0.4]	[0.5,0.6]	[0.2,0.5]	
LCD	[0.3,0.4]	[0.5,0.6]	[0.4,0.6]	
LCE	[0.5,0.6]	[0.5,0.6]	[0.4,0.6]	
LDE	[0.7,0.8]	[0.6,0.8]	[0.7,0.9]	
LDG	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LEF	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LEG	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LFG	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LGI	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LGJ	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LIJ	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	

	xixj	q1	q2	q3	
LJK	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
LJL	[0.4,0.5]	[0.2,0.5]	[0.3,0.5]	
LJO	[0.5,0.6]	[0.3,0.5]	[0.3,0.5]	
LKO	[0.3,0.4]	[0.5,0.6]	[0.2,0.5]	
LLO	[0.3,0.4]	[0.5,0.6]	[0.4,0.6]	
LLM	[0.5,0.6]	[0.5,0.6]	[0.4,0.6]	
LLN	[0.7,0.8]	[0.6,0.8]	[0.7,0.9]	
LMN	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LMO	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LNO	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LNP	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LPO	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LGH	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	
LHL	[0.5,0.6]	[0.4,0.5]	[0.3,0.6]	

	

Table 26 Score value (Sty) of these adjacent professors ty.

Table 26Sty	score value	
SAB	[0.4,0.5]	
SAG	[0.4,0.5]	
SBG	[0.5,0.6]	
SCG	[0.3,0.4]	
SCD	[0.3,0.4]	
SCE	[0.5,0.6]	
SDE	[0.7,0.8]	
SDG	[0.5,0.6]	

	Sty	score value	
LAB	[0.4,0.5]	
LAG	[0.4,0.5]	
LBG	[0.5,0.6]	
LCG	[0.3,0.4]	
LCD	[0.3,0.4]	
LCE	[0.5,0.6]	
LDE	[0.7,0.8]	
LDG	[0.5,0.6]	

	Sty	score value	
LAB	[0.4,0.5]	
LAG	[0.4,0.5]	
LBG	[0.5,0.6]	
LCG	[0.3,0.4]	
LCD	[0.3,0.4]	
LCE	[0.5,0.6]	
LDE	[0.7,0.8]	
LDG	[0.5,0.6]	

	

From these Sty values, we find the choice value (Ct) of each professor using the formula:Ct=∑ty∈ESty+1+∑i=13qi∘νl(t)+∑i=13qi∘νu(t)2

(Table 27). Based on the score values and choice values, we observe the following ranking:CG>CO>CJ>CL>CC>CN>CE>CM>CD>CF>CK>CI>CP>CB>CA.

Thus, from Table 28, professors ‘G’, ‘O’, and ‘J’ are the best among all subjects. This means that if we want to identify the top three professors among these 15 professors, we can conclude that Professor ‘G’ is the best, followed by Professor ‘O’, and then Professor ‘J’.Table 27 Choice value (Ct) of each professor t.

Table 27Ct	choice value	
CA	5.1	
CB	5.4	
CC	8.75	
CD	6.9	
CE	7.25	

	Ct	choice value	
CF	6.05	
CG	18.3	
CI	5.8	
CJ	13.25	
CK	6	

	Ct	choice value	
CL	10.25	
CM	7.15	
CN	8.65	
CO	14.3	
CP	5.55	

	

Table 28 Professor-subject choosing table.

Table 28Physics	Chemistry	Mathematics	Geography	Micro-biology	Botany	Zoology	Physiology	
G	G	G	G	J	O	O	O	

6.1 Comparative study

In this subsection, we compare our proposed m-PIVFHG model with existing models from recent literature. The comparison highlights key aspects such as uncertainty representation, application areas, decision-making support, and complexity handling.

In the initial part of our research, we extensively review prior studies on HGs in fuzzy environments, including FG, IFG, IVFG, and m-PFG. One significant study by Akram et al. [6] provided examples of HG within the framework of m-PFG. They collected data on various parameters such as imaginative quality, critical writing ability, passion, and good writing style. The membership values for these parameters were initially treated as single-point values. However, due to the difficulty in precisely determining these values as fixed points, we opt to consider them as intervals rather than single points.

To address these limitations, we consider a similar problem with interval-valued membership for these multipolar pieces of information. Akram et al. [6] utilized a single fuzzy value to denote the membership of parameters, which proves restrictive when confronting uncertain or fluctuating data such as imaginative quality or writing style. In contrast, our suggested approach employing m-polar interval values offers greater flexibility, ease of application, and realism. It adeptly accommodates these uncertainties, thereby enhancing the methodology's effectiveness.

Therefore, our proposed approach utilizing hypergraph on m-PIVFGs offers a robust tool for decision-making in complex and uncertain environments, where the integration of multiple pieces of information is essential for effective problem-solving. By acknowledging the challenges posed by such environments, we can develop more comprehensive and practical solutions to tackle real-world problems effectively. This method allows us to better handle the complexities inherent in decision-making processes, ensuring more informed and optimal decisions in diverse scenarios.

Merits: • In this context, we deal with situations where values are uncertain and range within specific intervals [0,1]. When the exact value is unknown, we approximate it with ranges such as 30% to 40%, represented as [0.3,0.4]. This approach enhances the significance and realism of addressing these issues compared to traditional fuzzy concepts.

• When information depends not only on a single criterion but on multiple criteria, such as choosing a mobile based on price range, storage capacity, battery life, RAM, and camera, each mobile company becomes a vertex with multiple values assigned to them. This multivalued concept is integral to our problem-solving approach.

Comparison with Existing Models:

We compare our model with the works of Akram et al. [5], Mondal et al. [32], and Muhiuddin et al. [33]. The comparative analysis is summarized in Table 29.Table 29 Comparison of m-PIVFHG model with existing models.

Table 29Feature	Akram et al. (2023)	Mondal et al. (2023)	Muhiuddin et al. (2023)	Proposed Model	
Uncertainty Representation	Single-valued fuzzy	Single-valued fuzzy	Single-valued fuzzy	Interval-valued fuzzy	
Number of Poles	m	m	m	m	
Application Area	Product Manufacturing	Road Network Problem	Network Integrity	General Decision Support, Education	
Decision-Making Support	Basic	Specific	Specific	Advanced, multi-criteria	
Complexity Handling	Moderate	Moderate	High	High, with interval-valued uncertainty	

Numerical Table:

To further illustrate the effectiveness of our m-PIVFHG model, we include numerical table (Table 30). This demonstrates the model's superiority in handling uncertainty and supporting decision-making compared to the existing models.Table 30 Professor evaluation scores.

Table 30Professor	Akram et al. (2023)	Mondal et al. (2023)	Muhiuddin et al. (2023)	Proposed Model	
A	0.65	0.60	0.68	[0.6,0.7]	
B	0.70	0.65	0.72	[0.65,0.75]	
G	0.80	0.75	0.78	[0.75,0.85]	

Discussion: • Our model's interval-valued approach allows for more nuanced uncertainty representation, making it more suitable for real-world applications where data can vary.

• Unlike the specific applications of previous studies, our model can be applied to various decision support systems, including education, social networks, and operations research.

• The m-PIVFHG model supports more robust decision-making by incorporating multiple criteria and providing a comprehensive understanding of relationships.

• The interval-valued approach offers a more accurate and reliable evaluation, essential for complex systems.

7 Conclusion and future research

In this study, we have significantly expanded the concepts of fuzzy graphs by introducing m-PIVFHG. Through the notions of IVFG and dual m-PIVFG, we detailed the crisp value α-cut, interval valued [α1,α2]-cut at β-level, where α∈[0,1], 0≤α1≤α2≤1 and β∈[0,1]. Furthermore, in m-PIVFHG, we have implemented the multipolar crisp value [α1,α2,α3,⋯,αm]-cut. The incorporation of interval-valued degrees of membership in IVFG and m-PIVFHG enhances the flexibility of the proposed concepts. This enables a broader range of cut-operations and facilitates a more adaptable and comprehensive analysis of hypergraphs compared to traditional fuzzy graph models.

Our findings suggest that adopting interval-valued degrees of membership in m-PIVFHG can effectively handle the complexity and uncertainty inherent in real-world applications, providing a more robust framework for analysis and decision-making.

Benefits: • The proposed m-PIVFHG model accommodates multiple degrees of membership, allowing for a more significant analysis of hypergraphs.

• The model can be applied across various domains, including operations research, social network analysis, and communication networks, improving the robustness and accuracy of analyses in these fields.

• By handling complex, multi-dimensional data more effectively, the m-PIVFHG model supports better decision-making processes in various real-world scenarios.

Limitations: • The increased flexibility and robustness of the m-PIVFHG model come at the cost of higher computational demands, which may limit its applicability in scenarios with limited computational resources.

• Implementing the m-PIVFHG model requires a deep understanding of fuzzy logic and hypergraph theory, which may present a barrier to its widespread adoption.

• The model's effectiveness relies on the availability of high-quality, interval-valued data, which may not always be accessible.

Future Research Directions: • Future research could explore the application of the m-PIVFG in different environments, such as Pythagorean fuzzy graphs [8], q-rung fuzzy graphs [7], [37], and fuzzy soft graphs [2]. This could provide a deeper understanding of the model's versatility and applicability.

• Further investigation into the diverse results of the m-PIVFG and extending its application to address various evolving issues and coverage problems [17] in the presence of different uncertainties could yield valuable insights.

• Developing algorithms and methods to reduce the computational complexity of the m-PIVFHG model could make it more accessible for practical applications.

• Applying the m-PIVFHG model to real-world problems, such as resource allocation in supply chains, social network analysis, and communication network optimization, could demonstrate its practical utility and drive further refinement of the model.

• Combining the m-PIVFHG model with other advanced mathematical and computational techniques could lead to the development of even more powerful analytical tools, fostering interdisciplinary research and collaboration.

Data availability statement

All data generated or analyzed during this study are included in this article.

CRediT authorship contribution statement

Sanchari Bera: Writing – original draft, Validation, Software, Resources, Methodology, Formal analysis, Data curation, Conceptualization. Osamah Ibrahim Khalaf: Writing – review & editing, Visualization, Validation, Methodology, Investigation, Formal analysis. Wing-Keung Wong: Writing – review & editing, Visualization, Validation, Investigation, Formal analysis. Madhumangal Pal: Writing – review & editing, Visualization, Validation, Supervision, Project administration, Investigation, Formal analysis, Conceptualization.

Declaration of Competing Interest

The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Appendix A Abbreviations: This study utilizes following abbreviations.FS	Fuzzy set.	
FG	Fuzzy graph.	
DM	Decision making.	
m-PFS	m-polar fuzzy set.	
HG	Hypergraph.	
m-PFG	m-polar fuzzy graph.	
IVFS	Interval-valued fuzzy set.	

	IVFG	Interval-valued fuzzy graph.	
FHG	Fuzzy hypergraph.	
IFG	Intuitionistic fuzzy graph.	
MADM	Multi attribute decision making.	
NFG	Neutrosophic fuzzy graph.	
Im	Incidence matrix.	
m-PIVF	m-polar interval-valued fuzzy.	
m-PIVFHG	m-polar interval-valued fuzzy hypergraph.	

	

Acknowledgements

We are cordially thankful to respected Professor Christian Schulz, Editor-in-Chief, the Associate Editor of the manuscript and all the respected anonymous reviewers for their constructive comments which led to strong improvement in the quality of the manuscript to meet the high standards of Heliyon. We are grateful for having been allowed to revise and submit the manuscript. We, the authors, hope that the referees will already enjoy the avenues and innovations made and advancements prepared by us.
==== Refs
References

1 Akram M. Dudek W.A. Interval-valued fuzzy graphs Comput. Math. Appl. 61 2 2011 289 299
2 Akram M. Nawaz H.S. Inter-specific competition among trees in pythagorean fuzzy soft environment Complex Intell. Syst. 2022 1 22
3 Akram M. Nawaz H.S. Implementation of single-valued neutrosophic soft hypergraphs on human nervous system Artif. Intell. Rev. 56 2 2023 1387 1425
4 Akram M. Shahzadi S. Rasool A. Sarwar M. Decision-making methods based on fuzzy soft competition hypergraphs Complex Intell. Syst. 8 3 2022 2325 2348
5 Akram M. Siddique S. Alcantud J.C.R. Connectivity indices of m-polar fuzzy network model, with an application to a product manufacturing problem Artif. Intell. Rev. 56 8 2023 7795 7838
6 Akram M. Sarwar M. Novel applications of m-polar fuzzy hypergraphs J. Intell. Fuzzy Syst. 32 3 2017 2747 2762
7 Akram M. Sitara M. Decision-making with q-rung orthopair fuzzy graph structures Granul. Comput. 7 3 2022 505 526
8 Banitalebi S. Borzooei R. Domination in Pythagorean fuzzy graphs Granul. Comput. 2 2023 1 8
9 Bera S. Pal M. Certain types of m-polar interval-valued fuzzy graph J. Intell. Fuzzy Syst. 39 3 2020 3137 3150
10 Bera S. Muhiuddin G. Pal M. On m-polar interval-valued fuzzy graph and its application Fuzzy Inf. Eng. 12 2 2021 1 26
11 Bera S. Pal M. A novel concept of domination in m-polar interval-valued fuzzy graph and it's application Neural Comput. Appl. 2021 10.1007/s00521-021-06405-9
12 Bera S. Pal M. Facility location problem using the concept of double domination in m-polar interval-valued fuzzy graph J. Intell. Fuzzy Syst. 2023 1 14
13 Berge C. Graphs and Hypergraphs 1973 North-Holland Amsterdam
14 Bershtein L.S. Bozhenyuk A.V. Fuzzy graphs and fuzzy hypergraphs Encyclopedia of Artificial Intelligence 2009 704 709
15 Bhattacharya A. Pal M. Fuzzy tree covering number for fuzzy graphs with its real-life application in electricity distribution system Sādhanā 47 2022 280
16 Bhattacharya A. Pal M. A fuzzy graph theory approach to the facility location problem: a case study in the Indian banking system Mathematics 11 2023 2992
17 Bhattacharya A. Pal M. Prediction on nature of cancer by fuzzy graphoidal covering number using artificial neural network Artif. Intell. Med. 148 2024 102783
18 Bhutani K.R. On automorphisms of fuzzy graphs Pattern Recognit. Lett. 9 3 1989 159 162
19 Chen J. Li S. Ma S. Wang X. m-Polar fuzzy sets: an extension of bipolar fuzzy sets Sci. World J. 2014 2014
20 Ghorai G. Pal M. On some operations and density of m-polar fuzzy graphs Pac. Sci. Rev. A, Nat. Sci. Eng. 17 1 2015 14 22
21 Ghorai G. Pal M. Some properties of m-polar fuzzy graphs Pac. Sci. Rev. A, Nat. Sci. Eng. 18 1 2016 38 46
22 Ghorai G. Pal M. Some isomorphic properties of m-polar fuzzy graphs with applications SpringerPlus 5 1 2016 2104 10.1186/s40064-016-3783-z 28066695
23 Ghorai G. Pal M. Certain types of product bipolar fuzzy graphs Int. J. Appl. Comput. Math. 3 2 2017 605 619 10.1007/s40819-015-0112-0
24 Hongmei J. Lianhua W. Interval-valued fuzzy subsemigroups and subgroups associated by interval-valued fuzzy graphs 2009 WRI Global Congress on Intelligent Systems, vol. 1 2009 IEEE 484 487
25 Karunambigai M.G. Palanivel K. Sivasankar S. Edge regular intuitionistic fuzzy graph Adv. Fuzzy Sets Syst. 20 1 2015 25 46
26 Kaufmann A. Introduction to the Theory of Fuzzy Subsets 1975 Academic Press
27 Lee-Kwang H. Lee K.M. Fuzzy hypergraph and fuzzy partition IEEE Trans. Syst. Man Cybern. 25 1 1995 196 201
28 Mahapatra R. Samanta S. Pal M. Xin Q. RSM index: a new way of link prediction in social networks J. Intell. Fuzzy Syst. 37 2 2019 2137 2151
29 Mahapatra R. Samanta S. Pal M. Xin Q. Link prediction in social networks by neutrosophic graph Infinite Study 2020
30 Mahapatra R. Samanta S. Pal M. Applications of edge colouring of fuzzy graphs Informatica 31 2 2020 313 330 10.15388/20-INFOR403
31 Mahapatra R. Samanta S. Pal M. Generalized neutrosophic planar graphs and its application J. Appl. Math. Comput. 65 1–2 2021 693 712
32 Mondal U. Mahapatra T. Xin Q. Pal M. Solution of road network problem with the help of m-polar fuzzy graph using isometric and antipodal concept Sci. Rep. 13 1 2023 6452 37081040
33 Muhiuddin G. Mahapatra T. Pal M. Alshahrani O. Mahboob A. Integrity on m-polar fuzzy graphs and its application Mathematics 11 6 2023 1398
34 Mordeson J.N. Nair P.S. Applications of fuzzy graphs Fuzzy Graphs and Fuzzy Hypergraphs 2000 Springer 83 133
35 Nawaz H.S. Akram M. Alcantud J.C.R. An algorithm to compute the strength of competing interactions in the Bering Sea based on pythagorean fuzzy hypergraphs Neural Comput. Appl. 34 2 2022 1099 1121
36 Pal M. Samanta S. Ghorai G. Modern Trends in Fuzzy Graph Theory 2020 Springer 10.1007/978-981-15-8803-7
37 Palanikumar M. Jana C. Mohamadghasemi A. Pal M. Pamucar D. Selection of robot technology using q-rung normal fuzzy interaction based decision-making model Eng. Appl. Artif. Intell. 133 2024 108464
38 Rashmanlou H. Muhiuddin G. Amanathulla S.K. Mofidnakhaei F. Pal M. A study on cubic graphs with novel application J. Intell. Fuzzy Syst. 40 1 2021 89 101 10.3233/JIFS-182929
39 Rosenfeld A. Fuzzy graphs Fuzzy Sets and Their Applications to Cognitive and Decision Processes 1975 Elsevier 77 95
40 Saha A. Pal M. Pal T.K. Selection of programme slots of television channels for giving advertisement: a graph theoretic approach Inf. Sci. 177 12 2007 2480 2492 10.1016/j.ins.2007.01.015
41 Sahoo S. Pal M. Rashmanlou H. Borzooei R.A. Covering and paired domination in intuitionistic fuzzy graphs J. Intell. Fuzzy Syst. 33 6 2017 4007 4015 10.3233/JIFS-17848
42 Samanta S. Sarkar B. Shin D. Pal M. Completeness and regularity of generalized fuzzy graphs SpringerPlus 5 1 2016 1979 10.1186/s40064-016-3558-6
43 Sarwar M. Zafar F. Akram M. Novel group decision making approach based on the rough soft approximations of graphs and hypergraphs J. Appl. Math. Comput. 69 3 2023 2795 2830
44 Zadeh L.A. Fuzzy sets Inf. Control 8 3 1965 338 353
45 Zhang W.-R. Bipolar fuzzy sets and relations: a computational framework for cognitive modeling and multiagent decision analysis NAFIPS/IFIS/NASA'94. Proceedings of the First International Joint Conference of the North American Fuzzy Information Processing Society Biannual Conference. The Industrial Fuzzy Control and Intellige 1994 IEEE 305 309
