1 Introduction

In 1965, Zadeh [34] first introduced the term fuzzy sets (FSs). It has become a useful tool to handle the problems with uncertainties. FSs assigns a degree of membership to each element in a given set S from [0, 1]. Due to its flexibility, various generalizations of FSs have been introduced. In this context, the concept of interval-valued fuzzy sets (IVFSs) was also explored by Zadeh [35]. The degree of membership in IVFSs was the sub-interval of [0, 1] instead of a single value from the interval [0, 1]. IVFSs has a large range for degree of membership and hence more flexible. Recently, numerous concepts such as (m,n)-Fuzzy sets [37], (a, b)-Fuzzy soft sets [38], T-spherical FSs [39] in the theory of FSs have been added. The concept of intuitionistic fuzzy sets (IFSs) was presented by Atanassov [2]. In IFSs, degree of membership (ω) and degree of non-membership (ν) were allocated to each member under the condition 0 ≤ ω + ν ≤ 1. However, if we assume the degree of membership is 0.5 and the degree of non-membership is 0.6, then we observe that the condition for IFSs violates. To overcome these difficulties, the notion of PyFSs was introduced by Yager [33]. In PyFSs, the condition for IFSs 0 ≤ ω + ν ≤ 1 was replaced with . Consequently, the above values have been adjusted. The more generalized form of IFSs termed interval-valued intuitionistic fuzzy sets (IVIFSs) was initiated by Atanassov [3]. In IVIFSs, every member has a degree of membership and degree of non-membership in the form of sub-intervals of [0, 1] like , respectively with the condition . For example, the degree of membership and degree of non-membership for IVIFSs are [0.1, 0.7] and [0.1, 0.2], respectively. However, if we consider the degree of non-membership [0.1, 0.6] instead of [0.1, 0.2] in IVIFSs, then again the condition for IVIFSs violates and we are unable to deal with such circumstances through IVIFSs. To handle such situations, the term interval-valued Pythagorean fuzzy sets (IVPyFSs) was proposed by Garg [11]. We observe in the above case of IVIFSs that the values [0.1, 0.7] and [0.1, 0.6] are problematic but have settled in the domain of IVPyFSs which exhibits the condition . On the other hand, if we consider the degree of non-membership [0.3, 0.8] instead of [0.1, 0.6] in IVPyFSs, then IVPyFSs fails to handle it. To deal with such obstacles, Peng et al. [20] introduced the concept of q-ROFSs, in which the sum of degree of membership (ω) and degree of non-membership (ν) such that also lies in [0, 1], where (q ≥ 3) . Consequently, q-ROFSs is comparatively more appropriate and adaptable for the uncertain data. Many researches have been conducted in the realm of q-ROFSs and its numerous applications were explored, such as the distance measure for q-ROFSs and its application in MCDM by Wang et al. [32], a q-ROFSs extension of the DEMATEL and its application in the education sector by Revalde et al. [26], the knowledge measure for the q-ROFSs and its application towards MADM introduced by Khan et al. [17] etc. As an extension of q-ROFSs, the concept of IVq-ROFSs was introduced by Joshi et al. [14]. IVq-ROFSs is most flexible to deal with complex real-world problems as compared to q-ROFSs. Due to its nature, numerous studies have been established on IVq-ROFSs, such as the new possibility degree measure for IVq-ROFSs and its application towards decision-making (DM) by Garg [9], a NGDM method based on CoCoSo and IVq-ROFSs and its application towards MAGDM by Zheng et al. [36], new exponential operation laws and operators for IVq-ROFSs and their application in group DM process by Grag [10]. Moreover, we can appropriately deal with the scenario described in the above example through IVq-ROFSs. By considering the degree of non-membership is [0.4, 0.8], the basic condition of IVq-ROFSs gives us a precise result i.e., , where (q ≥ 3) . In this way, we can handle many real-world problems by using IVq-ROFSs, more accurately. For more on IVq-ROFSs, one may consult [8].

On the other hand, the idea of fuzzy graphs (FGs) was proposed by Rosenfeld [25]. Similar to the case of FSs, we can easily model any relational phenomenon with uncertain information through FGs. Numerous terms have been described in the paradigm of FGs like fuzzy tolerance graphs by Pal et al. [22], fuzzy threshold graphs by Samanta et al. [28] etc. However, it has been observed that sometimes FGs become less effective and not applicable in dealing with several complex real-world problems. Consequently, numerous generalized forms of FGs have been explored in the literature. The very first generalization of FGs named interval-valued fuzzy graphs (IVFGs) was initiated by Akram et al. [1]. IVFGs is more effective and useful compared to the FGs, since in IVFGs we allocate subintervals of [0, 1] instead of numbers. Hence the domain of IVFGs is more flexible and has more capacity to deal with vagueness. Afterward, many notions in the setting of IVFGs were initiated like highly irregular IVFGs by Rashmanlou et al. [21] and balanced IVFGs by Rashmanlou et al. [24]. IVFGs is also a one-sided case because it deals only with the degree of membership. In this era, we occasionally encounter more complex systems with more options and have observed that IVFGs are not effective in such scenarios. Hence further extension of FGs termed intuitionistic fuzzy graphs (IFGs) was explored by Parvathi et al. [19], it comprises degree of membership and degree of non-membership. IFGs allows us to deal with belonging (degree of membership) and non-belonging (degree of non-membership) instead of FGs which deals only with belonging (degree of membership). Further to this, several new notions of IFGs were introduced by Shao et al. [29], some operations on IFGs were defined by Thilagavathi et al. [30] and some operations on IFGs via novel versions of the Sombor index for internet routing were introduced by Imran et al. [13]. Subsequently, many terms of crisp graphs (CGs) were shifted in IFGs. The concept of covering in IFGs was initiated by Sahoo et al. [27]. Further to this, the extension of IFGs called interval-valued intuitionistic fuzzy graphs (IVIFGs) was introduced by Atanassov [4]. IVIFGs also deals with the degree of membership and degree of non-membership and allocates the values as sub-intervals of [0, 1]. But to make the IFGs more effective and usable, the term Pythagorean fuzzy graphs (PyFGs) was introduced by Verma et al. [31]. To extend the domain of PyFGs, the term IVPyFGs was proposed by Akram et al. [7]. An extension of both IFGs and PyFGs termed q-ROFGs was introduced by Habib et al. [12], it has more capacity to deal with real-world problems with uncertainties. Many researches have been conducted on q-ROFGs and its numerous applications have been explored like the application of q-ROFGs in DM was described by Atheeque et al. [5], q-ROFGs structures and their application towards DM were investigated by Akram et al. [6] etc. Recently, Jan et al. [16] initiated the concept of interval-valued q-rung orthopair fuzzy graphs (IVq-ROFGs). In IVq-ROFGs, the degree of membership and degree of non-membership are expressed in the form of sub-intervals of [0, 1].

2 Literature review

FSs extends the concepts of classical sets to handle uncertainty and vagueness more precisely. FSs has proven a useful tool to solve the problems in various fields including artificial intelligence, control systems, decision-making, and data analysis. FSs assigns a degree of membership to each element in a given set S from [0, 1]. Due to its flexible nature, numerous extensions of FSs like IVFSs, IFSs, IVIFSs etc have been explored. IVFSs enables the representation of uncertainty and imprecision in a more flexible manner. IVFSs utilizes degree of membership as a subinterval of [0, 1] instead of a single value. Similarly, IFSs utilizes degree of membership and degree of non-membership as a single value from [0, 1]. IFSs provides us more comprehensive framework for handling vagueness. Furthermore, PyFSs is a cutting-edge extension of FSs which is more adaptable in handling uncertainties. PyFSs is based on the notion of Pythagorean membership grades, where the degree of membership and degree of non-membership satisfy the condition . Similarly, IVPyFSs is a significant addition in the theory of FSs, integrating the notion of IVFSs and PyFSs to effectively handle uncertainties. IVPyFSs assign interval-valued membership and non-membership degrees to each element. This innovative approach enables IVPyFSs to handle more complex sceneries that were not effectively handled by other generalizations of FSs like IFSs, PyFSs etc. Afterwards, the notion of q-ROFSs was established. q-ROFSs utilizes the degree of membership (ω) and degree of non-membership (ν) from [0, 1] satisfying the condition . A q-ROFSs is proven a cutting-edge extension of fuzzy set theory which provides a more flexible and effective framework for handling uncertainties. The model based on q-ROFSs has more capacity to deal such problems that were not dealt through traditional FSs like IFSs, PyFSs etc. Applications of q-ROFSs towards decision-making, image processing, data analysis, risk assessment, machine learning etc have been explored. By combining the concepts of IVFSs and q-ROFSs, the term IVq-ROFSs was introduced in the literature. IVq-ROFSs is the most flexible tool to deal with complex problems. IVq-ROFSs utilize degree of membership and degree of non-membership in the form of subintervals from [0, 1]. We can handle many real-world problems by using IVq-ROFSs more accurately. For more on IVq-ROFSs, one may consult [8].

FGs is the extension of classical graphs, incorporating the concepts of FSs to graphs. These models have more capacity to deal the problems containing uncertainties and vagueness. FGs assign fuzzy degrees of membership to vertices and edges, enabling the representation of imprecise relationships and uncertain connections. FGs has diverse applications including social network analysis, recommendation systems, image segmentation, data mining, network optimization etc. To make model of FGs more precise and flexible numerous terms have been introduced in the domain of FGs. One of the most important generalization of FGs named IVFGs was introduced by Akram et al. [1]. IVFGs assigns interval-valued membership degrees from [0, 1] to vertices and edges, respectively. Many concepts of classical graphs have been studied in the domain of different FGs. The terms covering and matching are fundamental concepts in the theory of graphs that have been discussed in the paradigms of different extensions of FGs, and their applications including operations research, computer science and social network analysis have been explored.

Recently, the concepts of covering and matching in FGs were discussed in [18]. Further extending the notion of FGs, the notion of IFGs was introduced by Parvathi et al. [19], it utilized degree of membership and degree of non-membership for nodes and edges from [0, 1]. Several new terms including covering and matching IFGs were introduced in [27]. Further extended form of IFGs termed PyFGs was introduced by Verma et al. [31]. These concepts were also discussed for some other generalizations of FGs (see [15,23]). One of the most important extension of both IFGs and PyFGs named q-ROFGs was introduced by Habib et al. [12], it has more capacity to deal with real-world problems with uncertainties. Many studies have been conducted on q-ROFGs and its numerous applications have been explored [5,6]. Recently, Jan et al. [16] introduced the notion of IVq-ROFGs. IVq-ROFGs allocates subintervals of [0, 1] as degree of membership and degree of non-membership for nodes and edges, respectively. Additionally, we elaborate the significance of study on IVq-ROFGs in Fig 1.

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Fig 1. Generalizations of the FSs and FGs.

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The abbreviations used in this manuscript are enlisted in Table 1.

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Table 1. Abbreviations table.

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Motivations: The concepts of covering and matching have been extensively discussed in classical graphs theory with many applications explored in different fields such as networking, data science, computer science etc have been explored. Recently, these concepts have been introduced in the paradigms of FGs, IFGs, PFGs, IVPFGs etc. The concepts of covering and matching in FGs have played a vital role in modeling the real-world problems with uncertainties. On the other hand, IVq-ROFGs has a unique vast structure with the capability to handle uncertainties more accurately and slight changes in degree of membership allow it to be adapt to other existing FGs models. Moreover, we observe that the notions of covering and matching have not yet been introduced in the domain of IVq-ROFGs. All of these factors motivated us to introduce these concepts in the realm of IVq-ROFGs and demonstrate their application.

Novelty of our work: We may describe the novelty of this work as follows.

1. We initiate the notions of SAC, SNC, SAC number, SIS and interval-valued q-rung orthopair fuzzy cycle (IVq-ROFC) by utilizing the SAs in the context of IVq-ROFGs and deduce many important results related to these terminologies.
2. The notions of complete IVq-ROFGs, complete bipartite IVq-ROFGs and cyclic IVq-ROFGs are introduced and investigated.
3. The concepts of matching, SM, PSM etc for IVq-ROFGs are explored.
4. To demonstrate the usefulness of the presented work, we provide an application towards the analysis of FACS with respect to the temperatures of different sensors. In this regard, we provide an algorithm which serves as a computational framework for FACS.
5. Finally, we conduct a comparative analysis of our presented model with the other existing models in the literature like IVIFGs, IVFGs and IVPyFGs.

Organization of this work: This manuscript is organized as: In Sect 2, we review the literature that is helpful for understanding the subsequent sections. Numerous useful terms related to FSs and FGs are presented in Sect 3. In Sect 4, we comprehensively discuss the notions of covering and matching in IVq-ROFGs. The terms like SNC, SAC and some useful notions like IVq-ROFC, SIS, SM and PSM based on SAs are also initiated. In Sect 5, we provide the application of covering in IVq-ROFGs towards the analysis of FACS. In Sect 6, we conduct a comparative analysis. We offer the conclusion of our study in Sect 7. We also provide the implications of our study including theoretical implications, practical implications, educational and industrial impacts etc. At the end, we provide a future work and directions of our study in Sect 8.

3 Preliminaries

In this section, we offer some helpful notions from the literature for further upcoming studies.

Definition 1. [34] A pair (ω, X) is a FSs on X, where ω : X → [0, 1] denotes the MF.

Definition 2. [35] An IVFSs A defined on U is expressed as , where and are L and U limits of degree of membership. Due to an interval for every m ∈ U.

Definition 3. [2] A set of the form of is called an IFSs, where denotes the degree of membership of s ∈ B, represents degree of non-membership of s ∈ B, with , for all s ∈ S.

Definition 4. [3] An IVIFSs S on U is the set , where : U → D([0, 1]), = ∈ D([0, 1]) : U → D([0, 1]), = ∈ D([0, 1]) and for all w ∈ U, .

Definition 5. [33] Let ω be the MF and ν be the NMF on universal set U. Then the PyFSs S is , where ω : V → [0, 1] and ν : V → [0, 1] such that , for all c ∈ V.

Definition 6. [33] A PyFSs R on U × U is called PyFR on U, described as , where and denote the degree of membership and degree of non-membership of xy in R, respectively such that , for all x, y ∈ U.

Definition 7. [20] A q-ROFR on set U can be described as

where and are the MF and NMF, respectively from the set U → [0, 1], where . The term is said to be an indeterminacy or hesitancy degree of element .

Definition 8. [11] An IVPyFSs on U can be described as ∈ V ⟩ with .

Definition 9. [14] An IVq-ROFSs over U is A = { x, (ω (x), ν (x)) } and ω and ν are mappings from U → D [0, 1], where and with , for

Definition 10. [25] A FGs G = (M, N) with the functions ν : P → [0, 1] and ω : P × P → [0, 1], where for a, b in P, ω (a, b) ≤ ν (a) ∧ ν (b) .

Definition 11. [1] Let be any graph. An IVFGs defined on is G = (M, N), where is an IVFSs on V and is an IVFSs on E such that : , for every (p, q) ∈ E.

Definition 12. [19] A graph of the form G = (M, N) is an IFGs, where

1. such that and denote the degree of membership and degree of non-membership of an element , respectively and
   (1)
   for all , (i= 1,2,3,...,n).
2. N ∈ M × M where and are such that
   (2)
   (3)
   and
   (4)
   for every , (i, j = 1, 2, 3,.. ., n).

Definition 13. [4] A pair H = (M, N) is an IVIFGs defined on = (V, E) such that M = , is said to be an IVIFSs on V and N = , be an IVIFSs on E ⊆ V × V, where and represent the lower degree of membership and upper degree of membership and and represents the lower degree of non-membership and upper degree of non-membership, respectively such that for all edges

,

,

satisfying , for all

Definition 14. [31] A PyFGs on U is represented by G = (V, S, R), where S is a PyFSs on U and R be the PyFR on U with

and , also and denote degree of membership and degree of non-membership of R, respectively.

Definition 15. [7] An IVPyFGs on U is a pair G = (S, R), where S is the IVPyFSs and R is a IVPyFR such that ; and , for all x, y ∈ U.

Definition 16. [12] A q-ROFGs on U is a pair , where G is a q-ROFSs on U and F is a q-ROFR on U × U such that

and , for all a, b ∈ U and q ≥ 1.

Definition 17. [16] A graph of the form is said to be IVq-ROFGs, if

1. such that , represent the lower degree of membership and upper degree of membership and , denote the lower degree of non-membership and upper degree of non-membership of , respectively where , for , for all , (i = 1, 2, 3,.. ., m)
2. , where and such that , where and , such that with , , for all

4 Covering and matching in IVq-ROFGs

In this portion, we introduce some useful definitions, results and the notions related to covering and matching in IVq-ROFGs.

Definition 18. The SAs of an IVq-ROFGs is an edge with

,

, .

where and are the lower degree of membership and lower degree of non-membership, respectively.

Now we introduce the notion of SNC in IVq-ROFGs with the help of SAs.

Definition 19. Let be an IVq-ROFGs. In an IVq-ROFGs , the collection of vertices Z that encompasses all SAs of is called SNC. An IVq-ROFW of SNC Z is describes as

where and denote the min of the lower degree of membership and upper degree of membership of all SAs and , represent the max of the lower degree of non–membership and upper degree of non–membership of all of SAs in an IVq–ROFGs that are incident to .

Definition 20. A set is a SNC number of IVq-ROFGs, if

where FW is the notation of fuzzy weight and the min degree of membership and max degree of non-membership of SNC is called a min SNC in IVq-ROFGs .

Definition 21. An IVq-ROFGs is a complete IVq-ROFGs, if

,

,

for all where

satisfying .

Definition 22. An IVq-ROFGs that can be partitioned into subsets and for or , , and , is said to be a bipartite IVq-ROFGs, if for all and ,

,

,

then is called a complete bipartite IVq-ROFGs.

Definition 23. An edge (arc) of the form with is said to be a weakest arc of IVq–ROFGs if is an edge with min and min .

Definition 24. A path S of lengths is a sequence of vertices , j = 1, 2, 3,.. ., s and the lower and upper degree of membership and lower and upper degree of non-membership of weakest arc is defined as its strength. If and s ≥ 3 then S is said to be a cycle and Ġ is called interval-valued q-rung orthopair fuzzy cycle (IVq-ROFC), if it contains more then one weakest edge (arc). The strength of a cycle is the strength of a weakest edge in it.

Theorem 25. Let be a complete IVq–ROFGs with number of nodes s. Then

where and denote lower degree of membership and upper degree of membership and , express the lower degree of non–membership and upper degree of non–membership of the least SAs in .

Proof: Let be a complete IVq-ROFGs. Then it is clear that each vertex associated to are associated through SAs. Thus either the collection (s − 1) of vertices is SNC of . Suppose a vertex in with min degree of membership and max degree of non–membership is such that is connected to the distinct vertices . Then from all the least SAs of , is the arc having degree of membership and degree of non–membership , where . Hence the set Z of vertices is the set forms SNC of with

where and denote the min of the lower degree of membership and upper degree of membership of SAs incident on , respectively. Then

where and denote the lower degree of membership and upper degree of membership of least SAs in IVq–ROFGs . Hence

Similarly

where and denote the max lower degree of non–membership and upper degree of non–membership of the SAs incident on , correspondingly. Then

where and denote the lower degree of non–membership and upper degree of non–membership of least SAs in IVq–ROFGs . Hence

□

Theorem 26. If a complete bipartite IVq-ROFGs Ǧ is divided into two subsets and , then

Proof: Let Ǧ be a complete bipartite IVq-ROFGs, then clearly all of the arcs of Ǧ are SAs. Assume that and are two subsets of the set of vertices of Ǧ such that every vertex in is connected to all vertices in . The collection of arcs of a complete bipartite graph Ǧ is the union of the collection of all arcs incident to each vertex of and the collection of all arcs incident to each vertex of . Moreover, , and their union are IVq–ROFGs in Ǧ. Obviously,

Hence

Similarly,

Also, we have

Hence

Similarly, we have

□

Proposition 27. If Ġ be an IVq–ROFC such that be a cycle with number of vertices in Ġ, then

Proof: In an IVq-ROFC Ġ, all of the arcs are SAs. Due to this, the no. of vertices in SNC are same in Ġ and . Now, the SNC number of is . Thus, the min no. of vertices in SNC of Ġ is . □

Definition 28. Let be an IVq-ROFGs. Then two nodes in are called SI, if there no SAs between these two nodes. If the collection of vertices have two vertices that are SI, then the collection (set) is called a SIS.

Definition 29. An IVq-ROFW of SIS Z in an IVq–ROFGs can be described as

where and denote the min of the lower degree of membership and upper degree of membership of the SAs and and represent the max of the lower degree of non–membership and upper degree of non–membership of the SAs of IVq–ROFGs , which are incident on .

SI number of IVq-ROFGs is represented by with

SIS having max degree of membership and min degree of non-membership is said to be an MSIS in IVq-ROFGs .

Proposition 30. If Ġ be an IVq–ROFC such that be a cycle with number of vertices in Ġ, then

is the SIS of vertices in Ġ with

is the SIS of vertices in Ġ with

is the SIS of vertices in Ġ with

is the SIS of vertices in Ġ with

Proof: In an IVq-ROFC Ġ, all of the arcs are SAs. Due to this, the no. of vertices in SIS of both Ġ and are same. Now, the SIS number of is . Thus, the max no. of vertices in SIS of Ġ is . □

Theorem 31. Let be a complete IVq-ROFGs. Then

where and denote the lower degree of membership and upper degree of membership and and represent the lower degree of non–membership and upper degree of non–membership of the least SAs in , respectively.

Proof: Assume that is a complete IVq-ROFGs. Then each of its arc is SAs and all of the vertices are linked to all other vertices in . Hence, is the only SIS for every . □

Theorem 32. If a complete bipartite IVq-ROFGs Ǧ is divided into two subsets and , then

Proof: Let Ǧ be a complete bipartite IVq-ROFGs. Then every arc in Ǧ is SA. Since each of the node in is linked to all other nodes in , and each of the vertex in is linked to every other vertex in . Therefore, and are said to be the SIS in Ǧ. □

Theorem 33. Let be an IVq-ROFGs without any isolated vertex. Then

Proof: Let min SNC of is . Then

and forms SIS of vertices. Moreover, contains the vertices which are incident on SAs of . Thus

(5)

Let MSIS of G^∘ is P₀ i.e., P₀ having the vertices which are not neighbor based on SAs. Thus V^◇ − P₀ having the vertices which cover all of the SAs of G^◇. Hence, V^◇ − P₀ is an SNC of G^∘, where and are the min lower degree of membership and upper degree of membership and , are the max lower degree of non-membership and upper degree of non-membership, respectively. Then

(6)

From (5) and (6), we have

Hence the proof is completed. □

Example 34. Consider an IVq-ROFGs provided in Fig 2. All of its arcs are SAs and all the SNC of IVq-ROFGs G^∘ are: , , , , , . Tables 2 and 3 denote the calculations regarding IVq-ROFW of SNC number . Hence .

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Fig 2. An IVq-ROFGs G^∘ for a SNC.

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Table 2. Determinations of the FW of SNC number.

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Table 3. FWs of SNC number.

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Example 35. Let us consider an IVq-ROFGs is strong demonstrated in Fig 3. The collection of SISs are: , are depicted in Table 4. Hence .

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Fig 3. A strong IVq-ROFGs G^∘ for a SIS.

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Table 4. FWs of SI number.

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Definition 36. Let G^∘ be a IVq-ROFGs without any isolated vertex. In IVq-ROFGs G^∘, an SAC is the set of SAs that covers all vertices of IVq-ROFGs G^∘. The FW of SAC Â is described as

where and denote the min of the lower degree of membership and upper degree of membership of all SAs and and represent the max of the lower degree of non-membership and upper degree of non-membership of SAs.

A set , is an SAC number of IVq-ROFGs, if

SAC with min degree of membership and max degree of non-membership is said to be a min SAC in IVq-ROFGs G^∘.

Proposition 37. If Ġ be an IVq–ROFC such that be a cycle with number of vertices in Ġ, then

Proof: In an IVq-ROFC Ġ, all of the arcs are SAs. Due to this, the no. of edges in SAC of both Ġ and are similar. also, the SAC number of is . Hence, is the min number of edges in SAC of Ġ. □

Theorem 38. Let G^∘ is a complete IVq-ROFGs. Then

in G^∘, represents the number of nodes.

Proof: Let G^∘ be a complete IVq-ROFGs. Then every arc of complete IVq-ROFGs is SAs and all vertices are connected to each other. Since all of the arcs are SAs in an IVq-ROFGs G^∘ and crisp graph Ġ. Then Ġ and G^∘ have the equal no. of edges. Now, is the SAC number of the crisp graph Ġ. Thus is the minimum numbers of arc in SAC of G^∘. □

Theorem 39. if a complete bipartite IVq-ROFGs Ǧ is partitioned into two subsets and . Then

is SAC in Ǧ with

is SAC in Ǧ with

is SAC in Ǧ with

is SAC in Ǧ with

Proof: Consider that Ǧ is a complete bipartite IVq-ROFGs. Then every arc of complete bipartite IVq-ROFGs is SAs and all of the nodes in are linked to all other nodes in . Since all of the arcs are SAs in IVq-ROFGs G^∘ and crisp graph Ġ, thus, both the graphs have same numbers of SAC. The SAC number of G^∘ is . Thus, the min number of arcs is is the SAC of Ǧ. □

Definition 40. Let be an IVq-ROFGs. The collection Å of SACs is called a SIS, if any two arcs do not share a vertex. The set Å is also called a SM in G^∘.

Definition 41. Let Å be an SM in IVq-ROFGs. If , then is said to be matched strongly to . The IVq-ROFW of SM Å can be described as

A collection is an SM in IVq-ROFGs, if

is the SM of

is the SM of

is the SM of

is the SM of

SM with max degree of membership and min degree of non-membership is called max SM in IVq-ROFGs G^∘.

Proposition 42. if Ġ be an IVq-ROFC such that be a cycle with number of vertices in Ġ, then

Proof: In an IVq-ROFC Ġ, all of the arcs are SAs. Due to this, the no. of edgs in SM of Ġ and are similer. Now, is the SM number of . Thus, is also the max number of edges in SM of Ġ. □

Theorem 43. Let G^∘ be a complete IVq-ROFGs. Then

is SM with

is SM with

is SM with

is SM with

where in G^∘, represents the number of nodes.

Proof: Let G^∘ be a complete IVq-ROFGs. Then, every arc of complete IVq-ROFGs is the SAs and all the nodes are connected to each other. Since all of the arcs are SAs in a crisp graph Ġ and IVq-ROFGs G^∘, Thus, in SM the graphs Ġ and G^∘ have equal numbers of arcs. In crisp graph Ġ the number of SM is represented by . Thus, the max number of arcs in SM of G^∘ is . □

Theorem 44. If a complete bipartite IVq-ROFGs Ǧ divided into subsets and , then

is SM in Ǧ with

is SM in Ǧ with

is SM in Ǧ with

is SM in Ǧ with

Proof: Let Ǧ be a complete bipartite IVq-ROFGs. Then every arc of complete bipartite IVq-ROFGs is the SAs and all of the vertices in are linked to every other vertices in . Since all of the arcs are SAs in IVq-ROFGs Ǧ and crisp graph Ġ, hence both of the graphs has equal amount of arcs in SM. Since is the SM number of Ǧ. Thus, the max number of arcs in SM of Ǧ is . □

Example 45. Refer to IVq-ROFGs given in Fig 4. Here, all of the arcs are SAs and all the SACs of IVq-ROFGs G^∘ are: , , , , , , , . The FW of SAC number of IVq-ROFGs G^∘ is calculated in Table 5. Hence, . Also, the only SM and SAC in G^∘ are these two sets and . Thus , . Thus, .

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Fig 4. An IVq-ROFGs G^∘ for SIAC.

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Table 5. Determinations of the FWs of SAC number.

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Example 46. Suppose an IVq-ROFGs G^∘ as shown in Fig 2. Clearly, ad, cd and bc are its SAs and all of the SACs of G^∘ are: , The IVq-ROFW of SAC number of IVq-ROFGs G^∘ is calculated in Table 6. Thus, . The set is the only SIAC. Hence,

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Table 6. Calculations of the FWs of SAC set.

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Definition 47. Let Å be an SM in IVq-ROFGs G^∘. Then, Å is called PSM, if all of the vertices of Å in G^∘ are strongly matched to some vertices of G^∘.

Example 48. Consider an IVq-ROFGs depicted in Fig 5. All of its arcs are SAs and the collections and are called PSM. All SM of G^∘ are: , , , . The IVq-ROFW of SM is calculated in Table 7. Hence, , .

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Fig 5. An IVq-ROFGs G^∘ for SM.

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Table 7. Calculations of the FWs of SM.

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5 Analysis of Fuzzy Air Conditioning System (FACS)

An air conditioner (AC) is an electrical device that ustilizes air conditioning system and helps to keep the temperature of the surroundings according to the need. The AC system starts when the thermostat detects that the indoor temperature is greater than the set (targeted) temperature. The compressor begins to circulate refrigerant that absorbs the heat from indoor air in the evaporator coil. The heated refrigerant gas is pumped towards the condenser coil, where it releases the heat in the outside air and condenses back into a liquid. Afterwards, the indoor air is circulated by the blower fan over the evaporator coil which cools and dehumidifies this air. Finally, the ductwork distributes the cool air into the room or building. The AC system turned off automatically as the set temperature on the thermostat was achieved. It cycles on again when the temperature rises above the set point.

Usually, air conditioning system depends on binary decisions (on/off) based on temperature. However, FACS works on fuzzy logic that allows more flexible and adaptable control. An air conditioning system is designed to maintain a particular temperature range while the FACS has the capability of obtaining precise and adaptable cooling responses based on varying conditions. FACS makes AC to maintain the indoor temperature by comparing the room temperature and the targeted temperature. However, sometimes we encounter variations in temperature at different points within a large hall. In such circumstances, we consider a hall divided in to sections (portions) to handle this complex situation. To detect temperature in different sections of a hall, we use the temperature sensors { a, b, c, d, e }. Sometimes, the temperature readings from two different sensors are either identical or interdependent, and the temperature in the area between the two sensors is influenced by both reading. In such cases, we model such types of complex situations using the concepts of covering and matching in IVq-ROFGs. Specifically, we address these circumstances using the concepts of SISs, SNC etc in IVq-ROFGs. We investigate the effectiveness or ineffectiveness of the temperature levels in comparison to the targeted temperature in different sections. Our aim is to identify the sections of the hall where the temperature levels are ineffective or fall outside the specified range. We evaluate these by utilizing the concept of SISs of covering in IVq-ROFGs. Let us consider that the temperature sensors { a, b, c, d, e } placed in different sections of the hall are represented by the vertices that detect temperature levels at different points while the edges denote the realtionship between two sensors as depicted in Fig 6. In IVq-ROFGs, membership degree of vertices and edges depict the effectiveness of the temperatures levels, while the non-membership degrees of vertices and edges depict their ineffectiveness. By effectiveness of the temperature level, we mean that the temperature is either close to the desired value or there is only a slight differences between hall’s temperature and the targeted temperature. Similarly, ineffectiveness of the temperature level indicates that the temperature does not meet our requirements or is completely unrelated. The degrees of membership and non-membership corresponding to each temperature sensor (vertex) are shown in Table 8 while the degrees of membership and non-membership of each edge are presented in Table 9. By utilizing the concept of MSIS of covering in IVq-ROFGs, we detect the effectiveness or ineffectiveness of the temperature levels in those sections of the hall where the current temperature doesn’t meet our requirements. The MSIS from the collection of SISs represents the sections where the temperature levels are nearly ineffective.

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Fig 6. An IVq-ROFGs of Temperature sensors.

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Table 8. The values for vertices of IVq-ROFGs of the Temperature sensors.

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Table 9. The values for edges of IVq-ROFGs of the Temperature sensors.

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Computational framework. First, we find the SISs and then we find the MSIS from the collection of SISs. By utilizing the proposed algorithm 1, we identify the SISs (see Table 10) of an IVq-ROFGs shown in Fig 6.

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Table 10. The collection of the SISs.

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Algorithm 1:

1. Suppose s be any vertex in X. Remove all the vertices adjacent to s.
2. Now suppose any other vertex in the remaining graph belongs to X. Hence different IS, including s can be obtained depending on the vertex selected from the remaining of the graph and in X.
3. For selection of all possible vertices repeat step 2.

Further to the above, MSIS is calculated on the basis of fuzzy weights (FWs), as the unique SISs are and . The IVq-ROFWs corresponding to the sets given above are and . Clearly, has max degree of membership and min degree of non-membership as compared to . This implies that is selected as MSISs from the Table 10 on the base of its FWs. The MSIS represents those temperature sensors whose detected temperature is totally different or independent. Furthermore, in between these two sections of a hall that have the temperature sensors b and d, we can see that there is a varying condition of temperature. In general, we can say that the temperature at sensor b is cold and the temperature at sensor d is hot. Then on the basis of this condition, we can adjust the FACS according to the need of such sections. In this regard, the following five fuzzy logic levels can be considered i.e., the temperature level is very hot (VH), is hot (H), is warm (W), is cold (C) and is very cold (VC). Basically, FACS maintains the temperature based on three commands i.e., heat (HE), cool (CO) and no change (NC). Fig 7 manipulates the FACS according to the fuzzy logic levels (i.e., VC, C, W, H, VH) and commands (i.e., HE, CO and NC). In Fig 8, the flow chart elaborates the working of FACS. Furthermore, the working of FACS for all fuzzy logic levels is presented in Table 11.

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Fig 7. FACS.

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Fig 8. Flow chart of the working of FACS.

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Table 11. A working of FACS.

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Our computational frame work comprises of algorithm 1 and pseudocode 1 establishes the facts about our proposed methodology. For further elaboration of our proposed model, we present two numerical examples in Sect 6 and verify this model is more useful to express the complex scenario compared to other models presented through IFGs and PyFGs.

Pseudocode 1:

1. Initialize Variables
   1. ◇ Create a list of sensors: sensors = [a, b, c, d, e]
   2. ◇ Define the target temperature range: target − temp = (20, 25)
   3. ◇ Input temperature readings: temp − readings = [30, 22, 15, 28, 40]
2. For Loop to Identify Sensors Outside the Target Range
   For (sensor = 0, sensor < 5, sensor + +) :
   If the temperature of the current sensor is outside the target range: Add the sensor and its temperature to outside-sensors.
3. Define Fuzzy Membership Functions for Temperature
   1. ◇ Define membership functions for temperature:
   2. ⋅ low for temperatures [0, 20]
   3. ⋅ ok for temperatures [20, 28]
   4. ⋅ high for temperatures [25, 50]
4. Define Fuzzy Membership Functions for Heating and Cooling
5. Define Fuzzy Control Rules
6. Create Fuzzy Control Systems
   1. ◇ Initialize cooling-ctrl for cooling logic.
   2. ◇ Initialize heating-ctrl for heating logic.
7. Process Each Sensor Using a For Loop
   For (sensor = 0, sensor < 5, sensor + +) :
   Retrieve the current temperature: temp − reading = temp − readings [sensor].
   If temp − reading < target − temp [0](temperatureisbelowrange) :
   
   1. ⋅ Apply the heating-ctrl control system.
   2. ⋅ Set the input to the sensor’s temperature.
   3. ⋅ Compute the heating output.
   4. ⋅ Display: Heating activated for sensor { sensors [sensor]} with output { heating − output }.
   Else if temp − reading > target − temp [1](temperatureisaboverange) :
   
   1. ⋅ Apply the cooling-ctrl control system.
   2. ⋅ Set the input to the sensor’s temperature.
   3. ⋅ Compute the cooling output.
   4. ⋅ Display: Cooling activated for sensor { sensors [sensor]} with output { cooling − output }.
   Else (temperature is within the target range):
   1. ⋅ Display: No change needed for sensor { sensors [sensor]}.

6 Comparative analysis and superiority of our proposed study

The notion of FGs was introduced by Rosenfeld. FGs use degree of membership for nodes and edges satisfying a condition. By utilizing FGs, many real-world vague problems were resolved and useful results were obtained. In graphs, the notion of covering and matching is very important and has numerous applications. In FGs, several vague problems have been resolved based on covering and matching. Several extensions of FGs were introduced to deals with uncertain problems. The most important extensions of FGs named IVFGs, IFGs and IVIFGs. FGs represents the data by using the degree of membership as a single element from [0, 1] and for IVFGs the degree of membership is in the form of sub-intervals of [0, 1]. However, with the passage of time real-world uncertain or vague problems become more complex that cannot be handled through FGs and IVFGs. Therefore, we need a model or a structure which is more flexible then previous one to obtain more accurate and useful outcomes. To tackle such kind of situations, the concept of IFGs was introduced. Let us consider an example that a person is agree or disagree to do work with an organization. The structure of FGs and IVFGs deal only with agree to do work and assign value of a membership from [0, 1] to conclude whether a person is agree or strongly agree. Sometimes we faced a negative opinion (disagree). To deal with both opinions (agree or disagree), we need a structure that has the ability to deal with such cases more precisely. In this regard, the notion of IFGs introduced in literature, IFGs utilizes the degree of membership and degree of non-membership as a single element from [0, 1] for nodes and edges. Here, we can deal with negative opinions (disagree) through the non-membership. With the passage of time, to handle complex problems more precisely, the concept of IVIFGs was introduced. IVIFGs is more accurate structure and has wide range compared to IFGs, FGs and IVFGs to make a decision. An IVIFGs uses degree of membership and degree of non-membership in the form of sub-intervals of [0, 1] for nodes and arcs. Subsequently, the notions of covering and matching in IVIFGs were introduced which become useful to deal complex problems and generating accurate conclusions for them. To prove the worthiness of our proposed study, we analyze our proposed notion of covering and matching in the domain of IVq-ROFGs with comparison to the notion of covering and matching in the domain of IVIFGs. In this regard, we deal an example by using covering in IVIFGs and IVq-ROFGs. For this, we consider the data set as shown in Table 12. We find the MSIS from the SISs of covering in the setting of IVIFGs and IVq-ROFGs. Furthermore, the comparison between the FWs of MSIS shows that the notion of covering and matching in IVq-ROFGs is better and more flexible than covering and matching in IVIFGs. Then we have the MSISs from the collection of SISs as:

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Table 12. The values of nodes in the domain of IVIFGs for analysis of FACS.

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Clearly, has the max degree of membership and min degree of non-membership as compared to . This implies that is our required MSIS of covering in IVIFGs. Furthermore, if we consider the data set as given in Table 8. An IVIFGs are unable to deal with data that given in Table 8. Because of the taken values for membership and non-membership violate the condition of IVIFGs, as shown in Example 49.

Example 49. Assume the value of VC cluster and of VC shown in Table 8, then by the condition

The above condition is not satisfied, hence we are unable to continue in the domain of IVIFGs.

To handle such situations, one of the most advanced extension of the FSs named q-ROFSs that allows a wide range for the degree of membership and the degree of non-membership was introduced. Furthermore, the concept of q-ROFGs was introduced by using q-ROFR. The q-ROFGs utilize the values of [0, 1] for the degree of membership and degree of non-membership of nodes and edges. Through the q-ROFGs we can obtain better results but these results are less appropriate and adaptable. Consequently, the concept of IVq-ROFGs was introduced in literature having the same structure as IVFGs, IVIFGs and for the degree of membership and degree of non-membership of the nodes and edges utilize the sub-intervals of [0, 1]. To make the structure of an IVq-ROFGs more efficient to deal with the big data set, we initiate the notion of covering and matching in IVq-ROFGs. We calculate the FWs of SNCs, SACs and the MSIS by utilizing the notion of covering in IVq-ROFGs. Hence, we can continue our work based on covering and matching in IVq-ROFGs by using the data set given in Table 8. Moreover, we also see in Example 50 that the data set provided in Table 8 can be modeled through IVq-ROFGs. However, we have observed that the IVFGs is unable to deal with the data set given in Table 8.

Example 50. Let us consider the values and of VC from the Table 8 with q = 3, then

Then the above conditions are satisfied and hence the notion of covering in IVq-ROFGs is more appropriate for our model such data.

Since MSISs are and . Then by utilizing the data set given in Table 8, we get the IVq-ROFW of the sets as and as:

Clearly, has max degree of membership and min degree of non-membership as compared to . This implies that is selected as the MSIS of covering in the domain of IVq-ROFGs. Moreover, Table 13 shows that the set has the max degree of membership and min degree of non-membership in both domains. However, in Table 12, we also notice that the MSIS of covering in the domain of IVq-ROFGs is more precise and effective than the MSIS of covering in the domain of IVIFGs. In Fig 9, we also provide a graphical comparison of calculated MSIS of covering in the frame of IVFGs and in the domain of IVq-ROFGs on the basis of FWs to validate that the notion of covering and matching is more effective in the realm of IVq-ROFGs to deal complex problems with a wide range of membership and non-membership degree from [0, 1]. At the end, we provide the characteristics comparison of IVq-ROFGs with IVFGs, IVIFGs and IVPyFGs in Table 14.

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Table 13. The MSISs for IVIFGs and IVq-ROFGs.

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Fig 9. Comparison of MSIS Z1 of the covering in IVFGs with MSIS Z1 of covering in IVq-ROFGs on the basis of calculated FWs.

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Table 14. The characteristic comparison of IVq-ROFGs with IVFGs, IVIFGs and IVPyFGs.

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Discussions. Finally, we provide the comparison of our application presented in Sect 5 through Table 13 and Fig 9. It is evident that if we utilize MSIS of covering in the setting of IVFGs instead of IVq-ROFGs, then we are unable to compare the temperature of sensors and their relation to targeted temperatures. Moreover, Fig 9 depicts that the findings are less valid in the case of IVIFGs and it may takes lot of time to settle down the temperature ranges. On the other hand, if we utilize the domain of IVq-ROFGs for our application, then we can easily analyze all the temperature sensors by using MSIS. Furthermore, Fig 9 shows that the range of the FW of MSIS in the domain of IVq-ROFGs is wide and flexible. In this regard, we can settle down properly a very high or very low temperature according to our need. Consequently, it is helpful to make an appropriate conclusions about temperatures on sensors and apply fuzzy logic to FACS in the paradigm of IVq-ROFGs.

7 Conclusion

There are several real-world uncertain problems that cannot be handled through FGs, IFGs and IVPyFGs, but can be explain easily through IVq-ROFGs due to its flexible nature. IVq-ROFGs offers more adaptability, accuracy and precision for the problems with uncertainties having two attribute yes or no. In this study, we have investigated the concepts of covering and matching based on SAs within the framework of IVq-ROFGs. We have added several novel terms like SM, SAC, SNC, SISs, PSM and produced many new results in the domain of IVq-ROFGs. Moreover, we have also addressed some special IVq-ROFGs like complete IVq-ROFGs, complete bipartite IVq-ROFGs etc. Overall, we have extended the concepts presented for IFGs, PyFGs etc. To show the usefulness of this work, an application of covering in IVq-ROFGs in the analysis of FACS based on temperature sensors is also provided. We have observed that applying covering in IVq-ROFG to temperature sensors offers a more flexible and adaptable framework for FACS, simplifying the management of indoor temperature control. This study also opens the door to extending these concepts to fuzzy soft and neutrosophic graphs to address other real-world problems.

8 Research limitations and future directions

Our study paves the way for significant advancements in interval-valued q-rung orthopair fuzzy graph theory and its applications, driving scientific progress and innovation in the years to come. In this work, we have presented the model based on limited temperature sensors. However, if we apply our model to large scale, it could be reformed by designing a particular algorithm to achieve the best solution. In this context, since the notion of interval-valued q-rung orthopair picture fuzzy graphs (IVq-ROPFGs) is not introduced in literature, one could provide comparatively more precise solution of the addressed problem by introducing the concepts of IVq-ROPFGs. Moreover, the algorithm presented in this study can be developed toward large-scale IVq-ROPFGs and the problems existing in several fields like block-chain, cyber-security, data science etc can be addressed.