q -Rung orthopair fuzzy directed hypergraphs: A new model with applications

Abstract

A q-rung orthopair fuzzy model is a powerful framework for illustrating uncertainty and vagueness. This model is more practical, flexible and appropriate as compared to fuzzy, intuitionistic fuzzy and Pythagorean fuzzy models. Directed hypergraphs are utilized to execute higher-order relationships in communications and social webbing. In this research study, we design a new framework for manipulating q-rung orthopair fuzzy data by means of combinative theory of q-rung orthopair fuzzy sets and directed hypergraphs. We introduce certain new concepts, including q-rung orthopair fuzzy directed hypergraphs, dual directed hypergraphs, line graphs and coloring of q-rung orthopair fuzzy directed hypergraphs. Further, we implement some interesting concepts of q-rung orthopair fuzzy directed hypergraphs to real life problems.

Keywords

dual directed hypergraphs algorithms

1 Introduction

A large number of characteristics are possessed by directed hypergraphs [6], which are considered as the basis of various techniques used in mathematics. This generalization of digraphs has broaden the applicability of graph theory and directed hypergraphs have been demonstrated to be more effective and competent modeling frameworks. A number of tactics, which are based on directed hypergraphs, are significant to be used for practical purposes in several environments. An extensive utilization of certain types of directed hypergraphs can be seen in computer science as a mathematical framework to represent the complex structures in various domains. Zadeh’s [26] fuzzy set has been acquired greater attention by researchers in a wide range of scientific areas, including management sciences, robotics, decision theory and many other disciplines. Jiang et al. [9] proposed some applications to multi-attribute decision-making based on covering variable precision (I, T)-fuzzy rough sets. Novel decision-making methods based on bipolar neutrosophic information were proposed by Zhan et al. [27]. Atanassov [5] originated the notion of intuitionistic fuzzy set (IFS), which is distinguished by a truth-membership (T) and falsity-membership (F) satisfying the requirement that the sum of both membership degrees should not exceed 1 and is considered as an extension of fuzzy sets. However, in many practical decision-making problems, the sum of truth-membership and falsity-membership may not be less than 1, but the sum of their squares may be less than 1. To handle such type of difficulties, Yager [23, 24] proposed the concept of Pythagorean fuzzy sets(PFSs), as an extension of IFSs. A PFS is characterized by a truth-membership and a falsity-membership and these two memberships in PFSs are needed to satisfy the condition T² (x) + F² (x) ≤1, which shows that PFSs have expanded space to assign orthopairs as compared to IFSs. PFSs can deal various real life problems more effectively but still there are the cases which cannot be handled using PFSs. Take an example, the truth-membership and falsity-membership values suggested by a decision maker are 0.8 and 0.9, respectively. Then, the problem can never be handled by the means of PFSs as 0 . 8² + 0 .9² = 1.45 > 1. In order to deal with such type of cases, Yager [25] introduced the idea of q-rung orthopair fuzzy sets(q-ROFSs). A q-ROFS is represented by means of two membership degrees, one is truth and the other is falsity, with the characteristic that the summation of qth power of truth-membership and qth power of falsity-membership should not be greater than 1. Thus, q-ROFSs extend the concepts of IFSs and PFSs, so that the facts and figures can be dealt in a widen range. After that, Liu and Wang [13] developed and applied certain simple weighted operators to aggregate q-ROFSs in decision-making. Certain applications of q-ROFSs in decision-making have been discussed in [7, 12]. Interval valued q-ROFSs and their properties were proposed by Joshi et al. [10]. Wang et al. [22] proposed some q-ROF Muirhead means with their application to multi-attribute group decision making.

The ambiguousness in the representation of different objects or in the relationships between them generates the essentiality of fuzzy graphs, which were originally studied by Rosenfeld [20] and further developed by Kaufmann [11] in 1977. Singh et al. [21] studied the knowledge representation using interval-valued fuzzy formal concept lattice. Pramanik et al. [19] introduced fuzzy φ-tolerance competition graphs. Zhan et al. [28] introduced novel decision-making algorithms based on intuitionistic fuzzy rough environment. Covering based multigranulation (I, T)-fuzzy rough set models and applications in multi-attribute group decision-making were proposed by Zhan et al. [29]. Recently, covering-based generalized IF rough sets with applications to multi-attribute decision-making and fuzzy β-covering based (I, T)-fuzzy rough set models and applications to multi-attribute decision-making have been studied in [30, 31]. Valuable contribution on fuzzy graphs and fuzzy hypergraphs is presented in [15]. Certain characteristics and problems corresponding to fuzzy coloring of fuzzy hypergraphs were studied by Goetschel [8]. Zhang et al. [32] proposed the covering-based general multigranulation intuitionistic fuzzy rough sets and corresponding applications to multi-attribute group decision-making.

Parvathi et al. [17] originated the notion of IF hypergraphs. Later on, this idea was generalized by Akram and Dudek [1]. Parvathi et al. [18] introduced the concept of IF directed hypergraphs. Coloring of IF directed hypergraphs and its certain properties were discussed by Myithili and Parvathi [16]. Akram and Luqman [2], [3] introduced certain concepts of bipolar fuzzy directed hypergraphs, single-valued neutrosophic directed hypergraphs and studied their certain properties. The same authors also developed a new decision-making method based on bipolar neutrosophic directed hypergraphs [4]. Recently, Luqman et al. [14] have defined q-ROF hypergraphs and studied their applications.

The main objective of this work is to apply the concept of q-ROFSs to directed hypergraphs to deal uncertain orthopairs in directed hypernetworks. The motivation of this work is the wide applicability of directed hypergraphs in the designing of large, dense and optical networks. They have a large applications in circuit clustering, system analysis and pattern recognition etc. IF digraphs(IFDGs) and IF directed hypergraphs(IFDHGs) have been used in modeling of various networks, including neural network models and gas pipeline networks etc. An IFDHG model is an effective framework for handling the uncertainty in those orthopairs, whose sum is not greater than 1. But IF and PF models are failed to deal with such type of problems in which a decision-maker assigns the orthopairs expressing the degree of satisfactory and unsatisfactory, whose sum or square sum is not less than 1. To widen the range of such applications, we define q-rung orthopair fuzzy directed hypergraphs, in which the sum of qth power of truth and falsity-membership degrees should be less than 1. In this way, a wide range of vague information can be expressed through this model and we can adjust the value of q to decide the data expression range, thus q-rung orthopair fuzzy directed hypergraphs are more suitable for the uncertain environment. Moreover, our proposed model is more general as compared to IF and PFDHG models, because when q = 1, this model reduces to an IFDHG model and when q = 2, it reduces to the PFDHG model. Being motivated by these advantages of q-ROF models, in this paper, we introduce certain new concepts, including q-rung orthopair fuzzy directed hypergraphs, dual directed hypergraphs, line graphs and coloring of q-rung orthopair fuzzy directed hypergraphs. Further, we implement some interesting concepts of q-rung orthopair fuzzy directed hypergraphs to real life problems.

This paper is arranged as: In section 2, q-rung orthopair fuzzy directed hypergraphs, dual directed hypergraphs and line graphs are discussed. We explain the methods for finding the line graphs and dual hypergraphs of q-rung orthopair fuzzy directed hypergraphs. Section 3 deals with the interesting concept of coloring of q-rung orthopair fuzzy directed hypergraphs. We define the novel concepts of p-coloring, $L -$ coloring and chromatic number. We also present some useful results which simplify the complex procedure of coloring. We reduce the coloring problem of a q-rung orthopair fuzzy directed hypergraph to crisp coloring by replacing it with a more simpler framework. Further, we define the spike reduction and sequentially simple q-rung orthopair fuzzy directed hypergraphs. In section 4, we discuss how the concept of q-rung orthopair fuzzy directed hypergraphs can be applied to analyze the real life problems. The last section deals with the conclusions and future directions.

2 q-Rung orthopair fuzzy directed hypergraphs

In this section, we define q-ROF digraphs and q-ROF directed hypergraphs. A q-ROF directed hypergraph generalizes the concept of IF directed hypergraph and broaden the space of orthopairs. We also define and construct the dual and line graphs of q-ROF directed hypergraphs. All these concepts are explained and justified through concrete examples.

Definition 2.1. [25] A q-rung orthopair fuzzy set (q-ROFS) Q in the universe X is an object having the representation Q={(x, T_Q(x), F_Q(x))|x ∈ X}, where the function T_Q : X → [0, 1] defines the truth-membership and F_Q : X → [0, 1] defines the falsity-membership of the element x ∈ X and for every x ∈ X, $0 \leq T_{Q}^{q} (x) + F_{Q}^{q} (x) \leq 1$ , q ≥ 1.

Furthermore, π_Q (x)= $\sqrt[q]{1 - T_{Q}^{q} (x) - F_{Q}^{q} (x)}$ is called a q-ROF index or indeterminacy degree of x to the set Q.

Definition 2.2. A q-rung orthopair fuzzy relation (q-ROFR) $R$ in X is defined as, $R$ ={〈x₁x₂, $T_{R} (x_{1} x_{2})$ , $F_{R} (x_{1} x_{2}) 〉 | x_{1} x_{2} \in X \times X},$ where $T_{R} : X \times X \to [0, 1]$ and $F_{R} : X \times X \to [0, 1]$ represent the truth-membership and falsity-membership functions of $R$ , respectively, such that $0 \leq T_{R}^{q} (x_{1} x_{2}) + F_{R}^{q} (x_{1} x_{2}) \leq 1$ , for all x₁x₂ ∈ X × X.

Definition 2.3. A q-rung orthopair fuzzy digraph on a non-empty set X is a pair $\vec{D} = (A, \vec{B})$ , where $A$ is a q-ROFS on X and $\vec{B}$ is a q-ROFR on X such that, $\begin{matrix} T_{\vec{B}} (x_{1} x_{2}) & \leq & min {T_{A} (x_{1}), T_{A} (x_{2})}, \\ F_{\vec{B}} (x_{1} x_{2}) & \leq & max {F_{A} (x_{1}), F_{A} (x_{2})}, \end{matrix}$ and $0 \leq T_{\vec{B}}^{q} (x_{1} x_{2}) + F_{\vec{B}}^{q} (x_{1} x_{2}) \leq 1$ , q ≥ 1, for all x₁, x₂ ∈ X .

Remark 2.1.

When q = 1, 1-ROF digraph is called an IF digraph.

When q = 2, 2-ROF digraph is called PF digraph.

Example 2.1. Let X={x₁, x₂, x₃, x₄} be the set of universe, $A$ ={(x₁, 0.7, 0.8), (x₂, 0.6, 0.9), (x₃, 0.5, 0.8), (x₄, 0.7, 0.8)} be a 5-ROFS and $\vec{B}$ be a 5-ROFR on X such that $0 \leq T_{\vec{B}}^{5} (x_{i} x_{j}) + F_{\vec{B}}^{5} (x_{i} x_{j}) \leq 1$ , for all x_i, x_j ∈ X . The corresponding 5-ROF digraph $\vec{D} = (A, \vec{B})$ is shown in Figure 1.

Fig. 1

5-ROF digraph $\vec{D}$ .

Definition 2.4. The support of a q-ROFS Q is defined as supp (Q)={x|T_Q (x) ≠0, F_Q (x) ≠1}. The height of a q-ROFS Q is defined as h*(Q)= $(max_{x \in X} T_{Q} (x), min_{x \in X} F_{Q} (x))$ . If h (Q)=(1, 0), then q-ROFS Q is called normal.

Example 2.2. Let Q={(q₁, 1, 0), (q₂, 0, 1), (q₃, 0.5,0.6), (q₄, 0.6, 0.7), (q₅, 0.9, 0.3)} be a 4-ROFS on X. Then, the support and height of Q are given as, supp (Q)={q₁, q₃, q₄, q₅} , h*(Q)=(1, 0), respectively. Note that, Q is normal.

Definition 2.5. A q-rung orthopair fuzzy directed hypergraph(q-ROFDH) $D$ on X is defined as an ordered pair $D = (Q, ξ)$ , where Q is the collection of q-ROF subsets of X and ξ is a family of q-ROF directed hyperedges (or hyperarcs) such that,

$\begin{matrix} T_{ξ} (E_{k}) & = & T_{ξ} (x_{1}, \dots, x_{m}) \\ \leq & min {T_{Q_{i}} (x_{1}), \dots, T_{Q_{i}} (x_{m})}, \\ F_{ξ} (E_{k}) & = & F_{ξ} (x_{1}, \dots, x_{m}) \\ \leq & max {F_{Q_{i}} (x_{1}), \dots, F_{Q_{i}} (x_{m})}, \end{matrix}$ for all x₁, x₂, ⋯, x_m ∈ X.

⋃_isupp (Q_i)=X, for all Q_i ∈ Q.

A q-ROF directed hyperedge ξ_i ∈ ξ is defined as an ordered pair (h (ξ_i) , t (ξ_i)), where h (ξ_i) and t (ξ_i) ∈ X - h (ξ_i), non-trivial subsets of X, are called the head of ξ_i and tail of ξ_i, respectively. A source vertex v in ξ_i is defined as h (ξ_i) ≠ v, for all ξ_i ∈ ξ and a destination vertex v′ in ξ_i is defined as t (ξ_i) ≠ v′, for all ξ_i ∈ ξ.

Definition 2.6. A q-ROFDH is called a backward q-ROFDH if all of its hyperarcs are B-arcs, i.e., ξ_i=(h (ξ_i) , t (ξ_i)) with |h (ξ_i) |=1, for all ξ_i ∈ ξ. A q-ROFDH is called a forward q-ROFDH if all of its hyperarcs are F-arcs, i.e., ξ_i=(h (ξ_i) , t (ξ_i)) with |t (ξ_i) |=1, for all ξ_i ∈ ξ.

Definition 2.7. The height of a q-ROFDH $D$ =(Q, ξ) is defined as $h^{★} (D)$ = {max(ξ_l) , min(ξ_m)}, where ξ_l=max T_{ξ
_j} (x_i) and ξ_m=min F_{ξ
_j} (x_i). Here, T_{ξ
_j} (x_i) and F_{ξ
_j} (x_i) denote the truth-membership and falsity-membership of vertex x_i to the directed hyperedge ξ_j, respectively.

Definition 2.8. Let $D$ =(Q, ξ) be a q-ROFDH. The order of $D$ , which is denoted by $O (D)$ , and is defined as $O (D)$ =∑_x∈X ∧ ξ_i (x). The size of $D$ , which is denoted by $S (D)$ , and is defined as $S (D)$ =∑_x∈X ∨ ξ_i (x).

Definition 2.9. A repeatedly occurring sequence v₁, ξ₁, v₂, ξ₂, ⋯, v_n-1, ξ_n-1, v_n of definite vertices and directed hyperarcs such that,

0 < T_ξ (ξ_i) ≤1 and 0 ≤ F_ξ (ξ_i) <1,

v_i-1, v_i ∈ ξ_i, i = 1, 2, 3, ⋯ , n,

is called a q-ROF directed hyperpath of length n - 1 from v₁ to v_n.

If v₁=v_n, then this q-ROF directed hyperpath is called a q-ROF directed hypercycle.

Definition 2.10. The strength of q-ROF directed hyperpath of length k, which connects the two vertices v₁ and v₂, is defined as, λ^k (v₁, v₂)={min {T_ξ (ξ₁), T_ξ (ξ₂), T_ξ (ξ₃), ⋯, T_ξ (ξ_k)}, max {F_ξ (ξ₁), F_ξ (ξ₂), F_ξ (ξ₃), ⋯, F_ξ (ξ_k)}}, v₁ ∈ ξ₁, v₂ ∈ ξ_k and ξ₁, ξ₂, ξ₃, ⋯, ξ_k are q-ROF directed hyperedges.

The strength of connectedness between v₁ and v₂ is given as, λ^∞ (v₁, v₂)= ${max_{k} T (λ^{k} (v_{1}, v_{2})), min_{k} F (λ^{k} (v_{1}, v_{2}))} .$

A connected q-ROFDH is one in which we have at least one q-ROF directed hyperpath between each pair of vertices of $D$ .

We now illustrate the Definitions 2.5, 2.6, 2.7, 2.8, 2.9 and 2.10 through an example of 5-ROFDH.

Example 2.3. Consider a 5-ROFDH $D$ =(Q, ξ), as shown in Figure 2.

Fig. 2

A 5-ROFDH.

In this 5-ROFDH, we have $\begin{matrix} ξ_{1} & = & {{(v_{1}, 0.8, 0.6), (v_{3}, 0.8, 0.5)}, {(v_{5}, 0.7, 0.8)}} \\ = & {t (ξ_{1}), h (ξ_{1})}, \\ ξ_{2} & = & {{(v_{1}, 0.8, 0.6), (v_{2}, 0.7, 0.9)}, \\ = & {(v_{3}, 0.8, 0.5), (v_{4}, 0.6, 0.8)}} = {t (ξ_{2}), h (ξ_{2})}, \\ ξ_{3} & = & {{(v_{3}, 0.8, 0.5), (v_{6}, 0.7, 0.6)}, {(v_{4}, 0.6, 0.8)}} \\ = & {t (ξ_{3}), h (ξ_{3})}, \\ ξ_{4} & = & {{(v_{4}, 0.6, 0.8), (v_{6}, 0.7, 0.6)}, {(v_{7}, 0.8, 0.7)}} \\ = & {t (ξ_{4}), h (ξ_{4})} . \end{matrix}$

A 5-ROF directed hyperpath from v₁ to v₇ of length 3 is shown through dashed lines and is given by an alternating sequence v₁, ξ₂, v₃, ξ₃, v₄, ξ₄, v₇ of distinct vertices and directed hyperarcs. The strength of this hyperpath is $\begin{matrix} λ^{3} (v_{1}, v_{7}) & = & {min {T_{ξ} (ξ_{2}), T_{ξ} (ξ_{3}), T_{ξ} (ξ_{4})}, \\ max {F_{ξ} (ξ_{2}), F_{ξ} (ξ_{3}), F_{ξ} (ξ_{4})}} \\ = & (0.6, 0.9), and \\ λ^{\infty} (v_{1}, v_{7}) & = & (0.6, 0.9) . \end{matrix}$

Note that, $D$ =(Q, ξ) is not connected because we don’t have a directed hyperpath between each pair of vertices, i.e., v₁ is not connected to v₆. A backward and forward 5-ROFDH is shown in Figures 3(a) and 3(b), respectively.

Fig. 3

Backward and Forward 5-ROFDHs.

Definition 2.11. A q-ROFDH $D$ =(Q, ξ) is linear if every pair of q-ROF directed hyperedges ξ_i, ξ_j ∈ ξ satisfies

supp (ξ_i) ⊆ supp (ξ_j) ⇒ i = j,

|supp (ξ_i) ∩ supp (ξ_j) |≤1.

Example 2.4. Consider a 5-ROFDH $D$ =(Q, ξ), as shown in Figure 2. In this 5-ROFDH, we have, supp (ξ₁)={v₁, v₃, v₅}, supp (ξ₂)={v₁, v₂, v₃, v₄} , supp (ξ₃)={v₃, v₆, v₄}, supp (ξ₄)={v₄, v₆, v₇} . Note that, supp (ξ_i) ⊆ supp (ξ_j) ⇒ i = j and $\begin{matrix} | supp (ξ_{1}) \cap supp (ξ_{2}) | & = & | {v_{1}, v_{3}} | = 2, \\ | supp (ξ_{1}) \cap supp (ξ_{3}) | & = & | {v_{3}} | = 1, \\ | supp (ξ_{1}) \cap supp (ξ_{4}) | & = & | {\emptyset} | = 0, \\ | supp (ξ_{2}) \cap supp (ξ_{3}) | & = & | {v_{4}, v_{3}} | = 2, \\ | supp (ξ_{2}) \cap supp (ξ_{4}) | & = & | {v_{4}} | = 1, \\ | supp (ξ_{3}) \cap supp (ξ_{4}) | & = & | {v_{4}, v_{6}} | = 2 . \end{matrix}$

That is, |supp (ξ_i) ∩ supp (ξ_j) |nleq1, for all ξ_i, ξ_j ∈ ξ. Hence, $D$ =(Q, ξ) is not linear.

Definition 2.12. Let $D$ =(Q, ξ) be a q-ROFDH. The q-rung orthopair fuzzy line graph of $D$ is the graph $l (D)$ =(X_l, ξ_l) such that,

X_l=ξ,

{ξ_i, ξ_j}∈ ξ_l ⇔ |supp (ξ_i) ∩ supp (ξ_j) | ≠ ∅, for i ≠ j.

The truth-membership and falsity-membership of vertices and edges of $l (D)$ are determined as follows:

X_l (ξ_i)=ξ (ξ_i),

T_{ξ
_l} ({ξ_i, ξ_j})=min {T_ξ (ξ_i) , T_ξ (ξ_j) |ξ_i, ξ_j ∈ ξ}, F_{ξ
_l} ({ξ_i, ξ_j})=max {F_ξ (ξ_i) , F_ξ (ξ_j) |ξ_i, ξ_j ∈ ξ}.

Theorem 2.1. Let $G$ =(U, ε) be a simple q-ROF directed graph. Then $G$ is the q-ROF line graph of a linear q-ROFDH.

Proof. Let $G$ =(U, ε) be a simple q-ROF directed graph. We suppose that $G$ =(U, ε) is connected, with no loss of generality. A q-ROFDH $D$ =(Q, ξ) can be formulated from $G$ as follows:

(i) The set of directed edges of $G$ will be taken as vertices of $D$ , i.e., ε={ε₁, ε₂, ε₃, ⋯ , ε_n} be the directed edges of $G$ and hence the set of vertices of $D$ . Let X={q₁, q₂, q₃, ⋯ , q_k} be the set of non-trivial q-ROFSs on U such that q_i (ε_j)=(1, 0), i=1, 2, 3, ⋯ , k, j=1, 2, 3, ⋯ , n.

(ii) Let U={u₁, u₂, u₃, ⋯ , u_j} then the directed hyperedges of $D$ are ξ={ξ₁, ξ₂, ξ₃, ⋯ , ξ_n}, where ξ_i are those directed edges of $G$ , which contain the vertex u_i as their incidence vertex, i.e., ξ_i={ε_j|u_i ∈ ε_j, j = 1, 2, 3, ⋯ , n}. Moreover, ξ (ξ_i)=U (u_i), i=1, 2, 3, ⋯ , k.

We now claim that $D$ =(Q, ξ) is linear q-ROFDH. Consider an arbitrary directed hyperedge ξ_j={ε₁, ε₂, ε₃, ⋯, ε_r} and from the defining relation of q-ROFDH, we have $\begin{matrix} T_{ξ} (ξ_{j}) & = & min {T_{q_{j}} (ε_{1}), T_{q_{j}} (ε_{2}), \dots, T_{q_{j}} (ε_{r})} \\ = & T_{U} (u_{i}) \leq 1, \\ F_{ξ} (ξ_{j}) & = & max {F_{q_{j}} (ε_{1}), F_{q_{j}} (ε_{2}), \dots, F_{q_{j}} (ε_{r})} \\ = & F_{U} (u_{i}) \geq 0, \end{matrix}$ i=1, 2, 3, ⋯ , k and ⋃_ksupp (q_k)=X, for all q_k.

We now prove that $D$ =(Q, ξ) is linear.

By our supposition, membership degree of each vertex ε_i of $D$ is (1, 0). Thus, we have supp (ξ_i) ⊆ supp (ξ_j) implies i = j.

Suppose on contrary that |supp (ξ_i) ∩ supp (ξ_j) |= {ε_l,ε_m}, i.e., these edges have two incidence vertices in common, which is contradiction to the fact that $G$ is simple. Hence, |supp (ξ_i) ∩ supp (ξ_j) |≤1, for 1 ≤ i, j ≤ r.

□

Theorem 2.2. A necessary and sufficient condition for $l (D)$ to be connected is that $D$ is connected.

Proof. Let $D$ =(Q, ξ) be a connected q-ROFDH and $l (D)$ =(X_l, ξ_l) be the line graph of $D$ . Suppose that ξ_i and ξ_j be two vertices of $l (D)$ and v_i ∈ ξ_i, v_j ∈ ξ_j, for v_i ≠ v_j. Since $D$ is connected then there exists an alternating sequence v_i, ξ_i, v_i+1, ξ_i+1, ⋯, ξ_j, v_j, which connects v_i and v_j. From the definition of strength of connectedness between v_i and v_j, we have $\begin{matrix} λ^{\infty} (ξ_{i}, ξ_{j}) & = & max_{k} T (λ^{k} (ξ_{i}, ξ_{j})), min_{k} F (λ^{k} (ξ_{i}, ξ_{j})) \\ = & {max_{k} (T_{ξ_{l}} (ξ_{i}, ξ_{i + 1}) \land T_{ξ_{l}} (ξ_{i + 1}, ξ_{i + 2}) \\ \land \dots \land T_{ξ_{l}} (ξ_{j - 1}, ξ_{j})), \\ min_{k} (F_{ξ_{l}} (ξ_{i}, ξ_{i + 1}) \lor F_{ξ_{l}} (ξ_{i + 1}, ξ_{i + 2}) \\ \lor \dots \lor F_{ξ_{l}} (ξ_{j - 1}, ξ_{j}))}, k = 1, 2, \dots \\ = & {max_{k} (T_{ξ_{l}} (ξ_{i}) \land T_{ξ_{l}} (ξ_{i + 1}) \land T_{ξ_{l}} (ξ_{i + 2}) \\ \land \dots \land T_{ξ_{l}} (ξ_{j - 1}) \land T_{ξ_{l}} (ξ_{j})), \\ min_{k} (F_{ξ_{l}} (ξ_{i}) \lor F_{ξ_{l}} (ξ_{i + 1}) \lor F_{ξ_{l}} (ξ_{i + 2}) \\ \lor \dots \lor F_{ξ_{l}} (ξ_{j - 1}) \lor F_{ξ_{l}} (ξ_{j}))}, \\ = & max T (λ^{k} (v_{i}, v_{j})), min F (λ^{k} (v_{i}, v_{j})) \\ = & λ^{\infty} (v_{i}, v_{j}) > 0 . \end{matrix}$

Hence, $l (D)$ is connected. By reversing the same procedure, we can easily prove that if $l (D)$ is connected then $D$ is connected. □

Let $D$ =(Q, ξ) be a q-ROFHG. The construction of a q-ROF directed line graph from a q-ROFDH is illustrated in Algorithm 1.

Algorithm 1

Input the number of directed hyperedges r of q-ROFDH $D$ =(Q, ξ).

Input the truth-membership and falsity membership of directed hyperedges ξ₁, ξ₂, ξ₃, ⋯, ξ_r.

Construct a q-rung orthopair fuzzy line graph $l (D)$ =(X_l, ξ_l), whose vertices are taken as the directed hyperedges ξ₁, ξ₂, ξ₃, ⋯, ξ_r.

Calculate the degrees of membership of vertices $l (D)$ =(X_l, ξ_l) as X_l (ξ_j)=ξ (ξ_j).

Draw an edge between ξ_i and ξ_j in $l (D)$ if |supp (ξ_i) ∩ supp (ξ_j) |≥1.

Calculate the degrees of membership of edges in $l (D)$ as, ξ_l (ξ_iξ_j)=(min {T_ξ (ξ_i), T_ξ (ξ_j)}, max {F_ξ (ξ_i), F_ξ (ξ_j)}) .

Definition 2.13. The 2-section graph of a q-ROFDH $D$ =(Q, ξ) is a q-ROFG $[D]_{2} = (X^{'}, E)$ such that,

(i) X=X′, i.e., the set of vertices of both graphs is same.

(ii) $E$ ={v_iv_j|v_i ≠ v_j, v_iv_j ∈ ξ_k,k =1,2,3, ⋯}, i.e., v_i and v_j are adjacent in $D$ .

We now justify the Definitions 2.12 and 2.13 through Example 2.5.

Example 2.5. Let $D$ =(Q, ξ) be a 7-ROFDH as shown in Figure 4. By following the above Algorithm, it’s line graph is constructed and shown by dashed lines.

Fig. 4

A 7-ROFDH and its line graph.

The 2-section graph of 7-ROFDH given in Figure 4 is shown in Figure 5.

Fig. 5

The 2-section graph of 7-ROFDH.

Definition 2.14. Let $D$ =(Q, ξ) be a q-ROFDH. The dual q-ROFDH $D^{d}$ =(X^d, ξ^d) of $D$ =(Q, ξ) is defined as,

(i) X^d=ξ is the q-ROF set of vertices of $D^{d}$ .

(ii) If |X|=n, then ξ^d is q-ROFS on the set of directed hyperedges {X₁, X₂, X₃, ⋯, X_n} such that X_i={ξ_j|v_i ∈ ξ_j, ξ_j ∈ ξ}, i.e., X_i is the set of those directed hyperedges in which v_i is a common vertex.

The membership degrees of X_i are defined as, $\begin{matrix} T_{ξ^{d}} (X_{i}) & = & min {T_{ξ} (ξ_{j}) | v_{i} \in ξ_{j}}, \\ F_{ξ^{d}} (X_{i}) & = & max {F_{ξ} (ξ_{j}) | v_{i} \in ξ_{j}} . \end{matrix}$

The method of forming the dual of q-ROFDH is described in Algorithm 2. We also explain this concept through an example.

Algorithm 2.

Let $D$ =(Q, ξ) be a q-ROFDH. The dual of $D$ is constructed as follows:

Input {v₁, v₂, v₃, ⋯, v_n} the set of vertices and {ξ₁, ξ₂, ξ₃, ⋯, ξ_m} the set of directed hyperedges of $D$ .

Formulate a q-ROFS of vertices of $D^{d}$ as X^d=ξ.

Define a mapping ψ : X → ξ, which maps the set of vertices to the directed hyperedges of $D$ , i.e., if vertex v_i is contained in ξ_l, ξ_l+1, ξ_l+2, ⋯, ξ_m then v_i is mapped onto ξ_l, ξ_l+1, ξ_l+2, ⋯, ξ_m.

Construct the directed hyperedges {X₁, X₂, X₃, ⋯, X_n} of $D^{d}$ such that X_i= {ξ_j|ψ (v_i) = ξ_j} .

Draw the q-ROF directed hyperedge, the vertex ξ_j of $D^{d}$ is associated to h (X_i) if and only if v_i ∈ t (ξ_j) in $D$ and viceversa.

Formulate the truth-membership and falsity-membership of directed hyperedges of $D^{d}$ as, $\begin{matrix} T_{ξ^{d}} (X_{i}) & = & min {T_{ξ} (ξ_{j}) | v_{i} \in ξ_{j}}, \\ F_{ξ^{d}} (X_{i}) & = & max {F_{ξ} (ξ_{j}) | v_{i} \in ξ_{j}} . \end{matrix}$

Example 2.6. Let $D$ =(Q, ξ) be a 7-ROFDH as shown in Figure 4. The dual 7-ROFDH $D^{d}$ =(X^d, ξ^d) of $D$ =(Q, ξ) is shown in Figure 6, which is constructed by following the Algorithm 2.

Fig. 6

Dual directed hypergraph of 7-ROFDH.

Theorem 2.3. The 2-section of dual of q-ROFDH $[D^{d}]_{2}$ is same as the line graph of $D$ , i.e., $[D^{d}]_{2}$ = $l (D)$ .

Proof. Let $D$ =(Q, ξ) be a q-ROFDH having {v₁, v₂, v₃, ⋯, v_n} the set of vertices and {ξ₁, ξ₂, ξ₃, ⋯, ξ_m} the set of directed hyperedges. Suppose that $l (D)$ =(X_l, ξ_l), $D^{d}$ =(X^d, ξ^d) and $[D^{d}]_{2}$ = $(X^{d}, E)$ be the line graph, dual directed hypergraph and 2-section of dual of $D$ , respectively. The 2-section $[D^{d}]_{2}$ has the same vertex set as that of $l (D)$ . Assume that the set of directed hyperedges of $D^{d}$ be {X₁, X₂, X₃, ⋯, X_n}. Obviously {ξ_iξ_j|ξ_i, ξ_j ∈ X_i} are the edges of $[D^{d}]_{2}$ and also the set of edges of $l (D)$ . We now show that ξ_l (ξ_iξ_j)= $E (ξ_{i} ξ_{j})$ . $\begin{matrix} ξ_{l} (ξ_{i} ξ_{j}) & = & (max {T_{ξ} (ξ_{i}), T_{ξ} (ξ_{i})}, \\ min {F_{ξ} (ξ_{i}), F_{ξ} (ξ_{i})}), \end{matrix}$ $\begin{matrix} = & (max {T_{ξ^{d}} (ξ_{i}), T_{ξ^{d}} (ξ_{i})}, \\ min {F_{ξ^{d}} (ξ_{i}), F_{ξ^{d}} (ξ_{i})}), \\ = & E (ξ_{i} ξ_{j}), \end{matrix}$ which completes the proof. □

We now justify the result of Theorem 2.3 through a concrete example.

Example 2.7. Let $D$ =(Q, ξ) be a 7-ROFDH as shown in Figure 4. Its line graph is constructed and shown by dashed lines in Figure 7.

Fig. 7

$l (D)$ .

The dual of $D$ is shown in Figure 6. We now determine the 2-section of $D^{d}$ , which is given in Figure 8.

Fig. 8

$[D^{d}]_{2}$ .

Thus, Figure 7 and Figure 8 show that $[D^{d}]_{2}$ = $l (D)$ .

3 Coloring of q-rung orthopair fuzzy directed hypergraphs

In this section, we define the (α, β)-level hypergraph of $D$ , which is a useful concept in coloring of q-ROFDHs. A sequence of real numbers, called the fundamental sequence of $D$ , is also defined using the (α, β)-level sets. The concept of fundamental sequence is used to prove various results related to the coloring of q-ROFDHs. Moreover, we define $L -$ coloring, chromatic number and p-coloring of $D$ . We also prove some useful results, which simplify the complicated procedure of coloring and finding the chromatic number of q-ROFDHs.

Definition 3.1. Let $D$ =(X, ξ) be a q-ROFDH. The (α, β)-level hypergraph of $D$ is defined as $D^{(α, β)}$ = (X^(α,β),ξ^(α,β)), where

ξ^(α,β)= ${ξ_{i}^{(α, β)} : ξ_{i} \in ξ}$ and $ξ_{i}^{(α, β)}$ ={x ∈ X|T_{ξ
_i} (x) ≥ α , F_{ξ
_i} (x) ≤ β} ,

X^(α,β)= $⋃_{ξ_{i} \in ξ} ξ_{i}^{(α, β)} .$

Definition 3.2. Let $D$ =(X, ξ) be a q-ROFDH and $D^{(α, β)}$ be the (α, β)-level hypergraph of $D$ . The sequence of real numbers ρ₁=(T_{ρ
₁}, F_{ρ
₁}), ρ₂=(T_{ρ
₂}, F_{ρ
₂}), ρ₃=(T_{ρ
₃}, F_{ρ
₃}),⋯,ρ_n=(T_{ρ
_n}, F_{ρ
_n}), 0 < T_{ρ
₁} < T_{ρ
₂} < T_{ρ
₃} < ⋯ < T_{ρ
_n}, F_{ρ
₁} > F_{ρ
₂} > F_{ρ
₃} > ⋯ > F_{ρ
_n} > 0, where (T_{ρ
_n}, F_{ρ
_n})= $h (H)$ such that

i. if ρ_i-1 =(T_{ρ
_i-1}, F_{ρ
_i-1}) < ρ=(T_ρ, F_ρ) ≤ ρ_i =(T_{ρ
_i}, F_{ρ
_i}) then ξ^ρ=ξ^{ρ
_i},

ii. ξ^{ρ
_i} ⊆ ξ^{ρ
_i+1},

is called the fundamental sequence of $D$ , denoted by f_S (D). The set of ρ_i-level hypergraphs ${D^{ρ_{1}}, D^{ρ_{2}}, D^{ρ_{3}}, \dots, D^{ρ_{n}}}$ is called the core hypergraphs of $D$ or simply the core set of $D$ and is denoted by $c (D)$ .

Definition 3.3. A q-ROFDH $D$ = (X, ξ) is ordered if $c (D)$ = ${D^{ρ_{1}}$ , $D^{ρ_{2}}$ , $D^{ρ_{3}}$ , ⋯, $D^{ρ_{n}}}$ is ordered, i.e., $D^{ρ_{1}} < D^{ρ_{2}} < D^{ρ_{3}}, \dots, D^{ρ_{n}}$ and is simply ordered if $c (D)$ is simply ordered.

Example 3.1. Consider a 2-ROFDH $D$ = (X, ξ), where X={x₁, x₂, x₃, x₄} and ξ={ξ₁, ξ₂, ξ₃} such that ξ₁={(x₁, 0.8, 0.1), (x₂, 0.8, 0.1)}, ξ₂={(x₁, 0.6, 0.2), (x₂, 0.6, 0.2), (x₃, 0.4, 0.3)}, ξ₃={(x₁, 0.4,0.3), (x₂, 0.4, 0.3), (x₄, 0.4, 0.3)}. By determining the (α, β)- level hypergraphs of $D$ , we have $D^{(0.8, 0.1)}$ = $D^{(0.6, 0.2)}$ and f_S (D)={(0.6, 0.2), (0.8, 0.1)} . Further, $D^{(0.4, 0.3)}$ = $D^{(0.6, 0.2)}$ . The corresponding sequence of level hypergraphs is shown in Figure 9.

Fig. 9

Fundamental sequence of $D$ .

We now define the primitive k-coloring (or simply a p-coloring), $L$ -coloring and chromatic number of q-ROFDHs and illustrate these concepts by considering a concrete example.

Definition 3.4. Let $D$ = (X, ξ) be a q-ROFDH. A primitive k-coloring C (or simply a p-coloring) is defined as a partition of X in k subgroups, called colors, such that the elements from at least two colors of C are contained in the support of every q-ROF directed hyperedge of $D$ .

Definition 3.5. Let $D$ = (X, ξ) be a q-ROFDH and $c (D)$ = ${D^{ρ_{1}}$ , $D^{ρ_{2}}$ , $D^{ρ_{3}}$ , ⋯, $D^{ρ_{n}}}$ be the set of core hypergraphs of $D$ . An $L$ -coloring is defined as a partition of X, with k components, into k subgroups {s₁, s₂, s₃, ⋯, s_k} such that C persuades a coloring for each core hypergraph $D^{ρ_{i}}$ =(X^{ρ
_i}, ξ^{ρ
_i}).

Remark 3.1. Note that, an $L$ -coloring of $D$ is a p-coloring, but in general, the converse does not hold. The preceding theorem states the condition under which an $L$ -coloring and p-coloring of $D$ coincides.

Theorem 3.1. Let $D$ = (X, ξ) be an ordered q-ROFDH and C is a p-coloring of $D$ then $L$ -coloring of $D$ is also C.

Definition 3.6. Let $D$ = (X, ξ) be a q-ROFDH and let k ≥ 2 be an integer then the k-coloring of vertex set is defined as a function κ:X → {1, 2, 3, ⋯ , k} such that for all $ρ \in f_{S} (D)$ and for each hyperedge ξ^ρ, which is not a loop, κ is not a constant on ξ^ρ.

The minimum integer k, for which there exists a k-coloring of $D$ is called chromatic number of $D$ , denoted by $χ (D)$ .

Example 3.2. Let $D$ = (X, ξ) be a 1-ROFDH, where X={t₁, t₂, t₃, t₄, t₅, t₆, t₇} and ξ={ξ₁, ξ₂, ξ₃, ξ₄, ξ₅, ξ₆, ξ₇} such that, $\begin{matrix} ξ_{1} & = & {(t_{1}, 0.6, 0.3), (t_{2}, 0.6, 0.3), (t_{4}, 0.5, 0.2)}, \\ ξ_{2} & = & {(t_{1}, 0.6, 0.3), (t_{3}, 0.6, 0.3), (t_{5}, 0.3, 0.1), \\ (t_{7}, 0.5, 0.2)}, \\ ξ_{3} & = & {(t_{1}, 0.6, 0.3), (t_{3}, 0.6, 0.3), (t_{6}, 0.2, 0.1), \\ (t_{7}, 0.5, 0.2)}, \\ ξ_{4} & = & {(t_{2}, 0.6, 0.3), (t_{3}, 0.6, 0.3), (t_{4}, 0.5, 0.2)}, \\ ξ_{5} & = & {(t_{2}, 0.6, 0.3), (t_{4}, 0.5, 0.2), (t_{5}, 0.3, 0.1), \\ (t_{7}, 0.5, 0.2)}, \\ ξ_{6} & = & {(t_{2}, 0.6, 0.3), (t_{4}, 0.5, 0.2), (t_{6}, 0.2, 0.1)}, \\ ξ_{7} & = & {(t_{4}, 0.5, 0.2), (t_{5}, 0.3, 0.1), (t_{6}, 0.2, 0.1)}, \end{matrix}$

Let ρ₁=(0.6, 0.3), ρ₂=(0.5, 0.2), ρ₃=(0.30.1) and ρ₄ = (0.2, 0.1). The corresponding ρ_i-level hyperedges are given as follows, $\begin{matrix} ξ^{ρ_{1}} & = & {{t_{1}, t_{2}}, {t_{1}, t_{3}}, {t_{2}, t_{3}}}, \\ ξ^{ρ_{2}} & = & {{t_{1}, t_{2}, t_{4}}, {t_{1}, t_{3}, t_{7}}, {t_{2}, t_{3}, t_{4}}, \\ {t_{2}, t_{7}, t_{4}}}, \\ ξ^{ρ_{3}} & = & {{t_{1}, t_{2}, t_{4}}, {t_{1}, t_{3}, t_{5}, t_{7}}, {t_{1}, t_{3}, t_{7}}, \\ {t_{2}, t_{3}, t_{4}}, {t_{2}, t_{4}, t_{5}}, {t_{2}, t_{4}}, {t_{4}, t_{5}}}, \\ ξ^{ρ_{3}} & = & {{t_{1}, t_{2}, t_{4}}, {t_{1}, t_{3}, t_{5}, t_{7}}, {t_{1}, t_{3}, t_{6}, t_{7}}, \\ {t_{2}, t_{3}, t_{4}}, {t_{2}, t_{4}, t_{5}, t_{7}}, {t_{2}, t_{4}, t_{6}}, \\ {t_{4}, t_{5}, t_{6}}} . \end{matrix}$

Suppose {C₁, C₂} is a coloring of $D^{ρ_{1}}$ . Then, {t₁, t₂}∩ {C₁, C₂} ≠ ∅, {t₁, t₃}∩ {C₁, C₂} ≠ ∅ and {t₂, t₃}∩ {C₁, C₂} ≠ ∅. Thus, C₁∩ C₂ ≠ ∅, which is a contradiction. Hence, $χ (D^{ρ_{1}})$ =3. {{t₁, t₂, t₃}, {t₄, t₅, t₆, t₇}} is the coloring of $D^{ρ_{2}}$ . Hence, $χ (D^{ρ_{2}})$ =2. Similarly, $χ (D^{ρ_{3}})$ =3 and $χ (D^{ρ_{4}})$ =3.

Definition 3.7. Let $D$ = (X, ξ) be a q-ROFDH and Q={q₁, q₂, q₃, ⋯, q_k} be the collection of non trivial q-ROFSs on X then Q is a q-rung orthopair fuzzy k-coloring if Q satisfies the following:

min {q_i, q_j}=(0, 1), if i ≠ j,

for every (α, β) ∈ (0, 1], $⋃_{i} q_{i}^{(α, β)}$ =X,

for every (α, β) ∈ (0, 1], each hyperedge $ξ_{j}^{(α, β)}$ possesses non-empty intersection with at least two color classes $q_{i}^{(α, β)}$ .

Observation 3.1. Let $D$ = (X, ξ) be a q-ROFDH having the fundamental sequence $f_{S} (D)$ ={ρ₁, ρ₂, ρ₃, ⋯, ρ_n}. Then the coloring of core hypergraph $D^{ρ_{i}}$ can be enlarged to the coloring of $D^{ρ_{i + 1}}$ if and only if a single color class of κ does not contain any hyperedge of $D^{ρ_{i + 1}}$ . Particularly, if $D$ is simply ordered then any coloring κ of $D^{ρ_{i}}$ may be elongated to the coloring of $D$ .

Theorem 3.2. Let $D$ = (X, ξ) be a q-ROFDH having the fundamental sequence $f_{S} (D)$ ={ρ₁, ρ₂, ρ₃, ⋯, ρ_n}. Let ${\tilde{D}}^{ρ_{n}}$ be the core coloring of $D^{ρ_{n}}$ then every coloring of $D^{ρ_{n}}$ is a coloring of $D$ if and only if for every $ρ \in f_{S} (D)$ there exists $A \in {\tilde{D}}^{ρ_{n}}$ such that $A \subseteq ξ_{i}^{ρ}$ , for each ξ_i ∈ ξ for which $ξ_{i}^{ρ}$ is a non loop edge.

Proof. Suppose the existance of some $ρ \in f_{S} (D)$ and ξ_i ∈ ξ such that $| ξ_{i}^{ρ} | \geq 2$ and $A ⊈ ξ_{i}^{ρ}$ , for every $A \in {\tilde{D}}^{ρ_{n}}$ . Let a color class is defined for the vertex set of $ξ_{i}^{ρ}$ . Construct a sub-hypergraph $D^{'}$ of $D$ , which is constructed by removing $ξ_{i}^{ρ}$ from the vertices of ${\tilde{D}}^{ρ_{n}}$ . Thus ${A \ ξ_{i}^{ρ} | A \in {\tilde{D}}^{ρ_{n}}}$ is the set of hyperedges of $D^{'}$ . Since every $ξ_{j}^{ρ_{n}} \in D^{ρ_{n}}$ , which is not a loop and also including $ξ_{i}^{ρ_{n}}$ , contains some $A \in {\tilde{D}}^{ρ_{n}}$ and this non loop edge $ξ_{j}^{ρ_{n}}$ has non empty intersection with the vertices of $D^{'}$ . Let {q₂, q₃, ⋯, q_k} be the coloring of $D^{'}$ then the coloring of $D^{ρ_{n}}$ is {ξ^ρ, q₂, q₃, ⋯, q_k}, where ξ^ρ is contained in single color class. Hence there exists a coloring of $D^{ρ_{n}}$ which is not a coloring of $D$ .

Conversely, assume that there exists some $ρ \in f_{S} (D)$ and ξ_i ∈ ξ such that $| ξ_{i}^{ρ} | \geq 2$ and $A \subseteq ξ_{i}^{ρ}$ , for every $A \in {\tilde{D}}^{ρ_{n}}$ . Suppose that ρ and ξ_i are taken as arbitrary but fixed and κ be the coloring of $D^{ρ_{n}}$ . Since κ is not a constant on A, it is also non constant on $ξ_{i}^{ρ}$ , hence κ is a coloring of $D$ .

The coloring problem of $D$ can be reduced to the correlated crisp coloring. It can be done by replacing $D$ with a more simpler framework $D^{Λ}$ , it will be noted that $D^{Λ}$ is ordered, simpler to color and every p-coloring of $D^{Λ}$ will generate the $L$ -coloring of $D$ .

Definition 3.8. A spike reduction of ξ_i ∈ P (X), which is denoted by $\tilde{ξ_{i}}$ , is defined as ${\tilde{ξ_{i}}}^{(α, β)} = {\begin{matrix} ξ_{i}^{(α, β)}, & if | ξ_{i}^{(α, β)} | \geq 2 \\ \emptyset, & if | ξ_{i}^{(α, β)} | \leq 1, \end{matrix}$ for 0 < α, β ≤ 1. Particularly, if ξ_i is a loop then $\tilde{ξ_{i}}$ =∅. □

Definition 3.9. Given $D$ =(Q, ξ) then $\tilde{D}$ = $(\tilde{X}, \tilde{ξ})$ , where $\tilde{ξ}$ = ${\tilde{ξ_{i}} | ξ_{i} \in ξ}$ .

Construction 3.1. Let $D$ = (X, ξ) be a q-ROFDH having the fundamental sequence $f_{S} (D)$ ={ρ₁, ρ₂, ρ₃, ⋯, ρ_n} and $c (D)$ = ${D^{ρ_{1}}$ , $D^{ρ_{2}}$ , $D^{ρ_{3}}$ , ⋯, $D^{ρ_{n}}}$ . Then the conversion of $D$ into $D^{s}$ is given in the following construction.

Obtain a partial hypergraph ${\bar{D}}^{ρ_{1}}$ of $D^{ρ_{1}}$ by abolishing all those directed hyperedges of $D^{ρ_{1}}$ that properly accommodate any other hyperedge of $D^{ρ_{1}}$ .

Subsequently, obtain a partial hypergraph ${\bar{D}}^{ρ_{2}}$ of $D^{ρ_{2}}$ by abolishing all those directed hyperedges of $D^{ρ_{2}}$ that properly accommodate any other hyperedge of $D^{ρ_{2}}$ or (properly or improperly) contain a hyperedge of partial hypergraph ${\bar{D}}^{ρ_{1}}$ . (It may be possible that ${\bar{D}}^{ρ_{2}}$ possesses no hyperedges, in such case existance of ${\bar{D}}^{ρ_{2}}$ is ignored.)

By following the same procedure, obtain a partial hypergraph ${\bar{D}}^{ρ_{3}}$ of $D^{ρ_{3}}$ by abolishing all those directed hyperedges of $D^{ρ_{3}}$ that properly accommodate any other hyperedge of $D^{ρ_{3}}$ or (properly or improperly) contain a hyperedge of partial hypergraph either ${\bar{D}}^{ρ_{1}}$ or ${\bar{D}}^{ρ_{2}}$ .

Following this iterative procedure, we obtain a subsequence of $f_{S} (D)$ , $ρ_{m}^{s} \dots < ρ_{1}^{s}$ =ρ₁ and the set of partial hypergraphs corresponding to this subsequence is $c (\bar{D})$ = ${{\bar{D}}^{ρ_{1}^{s}}$ , ${\bar{D}}^{ρ_{2}^{s}}$ , ${\bar{D}}^{ρ_{3}^{s}}$ , ⋯, ${\bar{D}}^{ρ_{m}^{s}}}$ from the $c (D)$ . It is obvious from above procedure that each ${\bar{D}}^{ρ_{i}^{s}}$ , 1 ≤ i ≤ m, contain non-empty set of hyperedges because all those hypergraphs having empty set of hyperedges have been eliminated from the consideration.

Construct the elementary q-ROFDH $D^{s}$ =(X^s, ξ^s) satisfying the following conditions:

$f_{S} (D^{s})$ = ${ρ_{1}^{s}$ , $ρ_{2}^{s}$ , $ρ_{3}^{s}$ , ⋯, $ρ_{m}^{s}},$

if ξ_j ∈ ξ^s then $h (ξ_{j}) \in {ρ_{1}^{s}$ , $ρ_{2}^{s}$ , $ρ_{3}^{s}$ , ⋯, $ρ_{m}^{s}},$

the family of hyperedges in ξ^s having heights $ρ_{k}^{s}$ is the collection of elementary q-ROFSs ${η (Q, ρ_{k}^{s}) | Q \in {\bar{D}}^{ρ_{k}^{s}}}$ , for all k, 1 ≤ k ≤ m.

Definition 3.10. Let $D^{Λ}$ be a q-ROFDH obtained from $\tilde{D}$ by the procedure described above, i.e., $D^{Λ}$ = $(\tilde{D})^{s}$ .

Definition 3.11. Let $D$ = (X, ξ) be a q-ROFDH having the fundamental sequence $f_{S} (D)$ ={ρ₁, ρ₂, ρ₃, ⋯, ρ_n} and $c (D)$ = ${D^{ρ_{1}}$ , $D^{ρ_{2}}$ , $D^{ρ_{3}}$ , ⋯, $D^{ρ_{n}}}$ with $D^{ρ_{i}}$ = $(X_{i}, E_{i})$ and the elements of $f_{S} (D)$ are ordered then $D$ is called sequentially simple if, whenever $E \in E_{i} \ E_{i - 1}$ then $E ⊈ X_{i - 1}$ , i = 1, 2, 3, ⋯ , n.

Theorem 3.3. Let $D$ = (X, ξ) be a sequentially simple q-ROFDH having core set $c (D)$ = ${D^{ρ_{i}}$ = $(X_{i}, E_{i}) | i$ =1, 2, 3, ⋯ , n} and the elements of $f_{S} (D)$ are ordered. Suppose that $E \in E_{j + k} \ E_{j}$ , j < n and k ∈ {1, 2, 3, ⋯ , n - j} then $E ⊈ X_{j}$ .

Proof. The general proof of this theorem is illustrated by considering an example. Assume that $E \in E_{j + 3} \ E_{j}$ , then

(i) either $E \in E_{j + 2}$ or $E \notin E_{j + 2}$ . In the succeeding condition $E \in E_{j + 3} \ E_{j + 2}$ , which indicates that $E ⊈ X_{j + 2}$ , thus $E ⊈ X_{j}$ because X_j ⊆ X_j+2. Now suppose that $E \in E_{j + 2}$ . Then

(ii) either $E \in E_{j + 1}$ or $E \notin E_{j + 1}$ . In the succeeding condition $E \in E_{j + 2} \ E_{j + 1}$ , which indicates that $E ⊈ X_{j + 1}$ , thus $E ⊈ X_{j}$ because X_j ⊆ X_j+1. Now suppose that $E \in E_{j + 1}$ . Then

(iii) since $E \notin E_{j}$ , this implies that $E \in E_{j + 1} \ E_{j}$ . Thus $E ⊈ X_{j}$ . hence it is clear that $E ⊈ X_{j}$ .

□

Theorem 3.4. Let $D$ = (X, ξ) be a sequentially simple q-ROFDH the $\tilde{D}$ , $D^{s}$ and $D^{Λ}$ are also sequentially simple q-ROFDHs.

Proof. Since $D$ = (X, ξ) is a sequentially simple q-ROFDH. Since $\tilde{D}$ is obtained by removing all those hyperedges of $D$ , which are spikes(loops) and also by eliminating all terminal spikes from the directed hyperedges of $D$ . Certainly $\tilde{D}$ is a sequentially simple q-ROFDH. Also the skeleton of $D$ , denoted by $D^{s}$ , is a sequentially simple q-ROFDH. Therefore, $D^{Λ}$ = $(\tilde{D})^{s}$ is also sequentially simple q-ROFDH. □

4 Applications

4.1 The most proficient arrangement for hazardous chemicals

Hazardous waste is a type of waste that is considered to have potential and substantial threats to environment and human health. There are many human activities, including medical practice, industrial manufacturing procedures and batteries that generate the hazardous waste in various categories, including solids, gases, liquids and sludges. The improper arrangement of these hazardous wastes results in many serious tragedies. Serious health issues, including cancer, birth defects and nerve damage may occur due to improper handling for those who ingest the contaminated air, water or food. Remediation and cleanup cost of these hazardous substances may amount to millions and billions of dollars. To ensure the well being of population, protection of surrounding environment and to avoid any type of threat or hazard, proper management of hazardous chemicals is extremely important. Consider a 5-ROFDH model to illustrate the arrangements of some compatible and incompatible elements. The connections in this model describe the management system of these elements,i.e., only those chemicals are connected together through an hyperedge, which have some compatibility. A q-ROFDH can be used to well demonstrate the management system of hazardous elements. The 5-ROFDH model of some compatible and incompatible elements is shown in Figure 10. The set of oval vertices G={G₁, G₂, G₃, G₄, G₅} of this directed hypergraph represents the types of those elements, which are adjacent to them. The description of these vertices is given in Table 1. For the cost efficient and secure management of hazardous elements, it is imperative to fill the containers up to 75% and also the container’s material should be compatible to the elements stored in it. Only those chemical substances are connected through the same directed hyperedges, which are compatible to each other and are not dangerous when stored together. For a proficient management of such elements, one should know the characteristics of hazardous elements such as corrosivity, reactivity or toxicity of these elements. A 5-ROFS Q describes the corrosivity of these chemical substances.

Fig. 10

The arrangement for hazardous elements using a 5-ROFDH model.

Table 1

Description of oval vertices

	Category	Membership values	Proficiency	Ineptness
GROUP1	Inorganic Acids	(0.7, 0.3)	70%	30%
GROUP2	Organic Acids	(0.72, 0.28)	72%	28%
GROUP3	Caustics	(0.71, 0.29)	71%	29%
GROUP4	Amines and Alkanolamines	(0.75, 0.25)	75%	25%
GROUP5	Halogenated Compounds	(0.73, 0.27)	73%	27%

$\begin{matrix} Q & = & {(w_{1}, 0.81, 0.23), (w_{2}, 0.81, 0.23), \\ (w_{3}, 0.81, 0.23), (w_{4}, 0.90, 0.17), \\ (w_{5}, 0.90, 0.17), (w_{6}, 0.90, 0.17), \\ (w_{7}, 0.87, 0.13), (w_{8}, 0.87, 0.13), \\ (w_{9}, 0.87, 0.13), (w_{10}, 0.75, 0.30), \\ (w_{11}, 0.70, 0.20), (w_{12}, 0.85, 0.20), \\ (w_{13}, 0.70, 0.10), (w_{14}, 0.70, 0.10), \\ (w_{15}, 0.90, 0.20)} . \end{matrix}$

Table 2 describes the importance of defining this 5-ROFS. The containers which are holding these chemicals should be in good condition, non leaking and compatible and these wastes should not be kept in a container that is made of an incompatible material. For example, acids must not be stored in metal material, hydrofluoric acid should not be stored in glass and lightweight polyethylene containers should not be used to store or transfer solvents. Thus, one should make sure that containers possess high level of compatibility to chemicals. We now consider a set of containers/cabinets C={C₁, C₂, C₃, C₄, C₅} and define five 5-ROFSs on C according to their compatibility with these elements. For example, the membership degrees C₁ (G₁)=(0.001, 0.980) implies that C₁ container is made up of such material which is incompatible to store inorganic acids and suitable to store organic acids as C₁ (G₂)=(0.81, 0.23). Similarly, by taking the same assumptions, we define other 5-ROFSs as given in Table 3. We have followed the (max, min) strategy to find out the compatibility of containers to chemicals. For example, we note that the container C₁ is made up of such material whose compatibility with different groups of chemicals can be checked through Table 3. The maximum truth degree and minimum falsity degree corresponding to inorganic acids concludes that C₁ container is compatible to store these chemicals. Similarly, it can be noted from Table 3 that organic acids should be stored in C₂ container as this is highly compatible to inorganic acids, so this storage will be most secure and risk less. Note that, the material of C₂ is compatible to organic acids, caustics and halogenated compounds but we will use this container to store organic acids because truth-membership degree is greatest in this case. In the same way, we find that C₃ is good for halogenated compounds, C₄ is used to store amines and alkanolamines and C₅ is suitable for storing caustics. The graphical representations of these storages is shown in Figure 11. We have proposed the simplest method to arrange the chemical substances and to store them safely. Thus, by taking the above model under consideration, hazardous chemicals can be systemized in a more appropriate and acceptable manner to reduce the precarious risks to human health and environment.

Table 2

Corrosivity and Fortifying level of square vertices

Square vertices	Corrosivity	Vitriolicity	Square vertices	Corrosivity	Vitriolicity
Nitric acid	81%	23%	Triethylamine	75%	30%
Sulfuric acid	81%	23%	Diethanolamine	70%	20%
Hydrochloric acid	81%	23%	Ethylenediamine	85%	20%
Acetic acid	90%	17%	Chlorobenzene	70%	10%
Butyric acid	90%	17%	Trichloroethylene	70%	10%
Formic acid	90%	17%	Trichlorofluoromethane	90%	20%
Sodium hydroxide	87%	13%	Solutions	87%	13%
Ammonium hydroxide	87%	13%

Table 3

Compatibility and incompatibility levels of containers to chemicals

C	Inorganic	Organic	Caustics	Alkanolamines	Compounds
C ₁	(0.81, 0.23)	(0.001, 0.980)	(0.10, 0.75)	(0.001, 0.980)	(0.010, 0.908)
C ₂	(0.10, 0.83)	(0.90, 0.17)	(0.75, 0.10)	(0.010, 0.908)	(0.75, 0.10)
C ₃	(0.001, 0.980)	(0.81, 0.23)	(0.10, 0.83)	(0.10, 0.83)	(0.91, 0.23)
C ₄	(0.10, 0.83)	(0.81, 0.23)	(0.90, 0.17)	(0.81, 0.23)	(0.81, 0.23)
C ₅	(0.001, 0.980)	(0.71, 0.23)	(0.930, 0.200)	(0.001, 0.980)	(0.870, 0.210)

Fig. 11

Graphical representations of storages of chemical substances.

4.2 Assessment of collaborative enterprise to achieve a particular objective

Collaboration is the demonstration of working as a team of members to achieve some piece of work, including research projects. Many organizations are realizing the significance of collaboration as a key factor to innovations. The collaborative work provides more opportunities for studying team-work skills and improves the personal and professional relationships. Here, we consider a few projects in chemical industry, which are assigned to different groups of trainees. A 7-ROFDH model is used to well demonstrate this collaborative activity of different teams/groups.

4.2.1 The project possessesing the powerful collaboration:

Consider the peculiar projects in the field of chemical industry, including Zero Energy Homes, Heat Exchanger Network Retrofit, Genetic Algorithms for Process Optimization, Progressive Crude Distillation, Water Management (for pollution prevention) and Design of LNG Facilities. The assignment of these projects to different groups is well explained through a 7-ROFDH as shown in Figure 12.

Fig. 12

A 7-ROFDH to represent the assigned projects to trainees.

Note that, the set of triangular vertices {p₁, p₂, p₃, p₄, p₅, p₆} represents the projects that are considered to be worked on and the set of circular vertices {t₁, t₂, t₃, t₄, t₅, t₆, t₇, t₈, t₉, t₁₀, t₁₁, t₁₂} represents the trainees, to whom these projects are assigned. Each directed hyperedge connects the corresponding project to it’s allocated trainees. The projects assigned to different groups are illustrated through Table 4. Note that, collaborative competency levels of different teams narrate that how much mutual understanding is there between the members of corresponding teams towards their projects. For example, the trainees of “Zero Energy Homes" project have 65% collaborative competency, i.e., they give respect to each other’s ideas, contribution and acknowledge the opinions of other trainees and their collective strength to achieve the goal is 65%. Incompetency degree shows that they have 11% conflicts of ideas and opinions. Similarly, the collaborative competency of all other teams can be studied through the table. Now, to evaluate the strength of determination and competent behavior of all teams towards their collaborative project, we calculate the heights of all directed hyperedges, which are given in Table 5. The directed hyperedge having maximum height, i.e., maximum truth-membership and minimum falsity-membership will correspond to the most efficient team working in collaboration. Note that, ξ₅ and ξ₆ have maximum heights showing that {t₇, t₉, t₁₂} and {t₁₀, t₁₂} share the most powerful collaborative characteristics. The method adopted in this part can be explained by a simple algorithm given in Table 6.

Table 4

Collaboration capabilities of groups to projects

Assigned Projects	Collaboration Team	Collaborative Competency	Collaborative Incompetency
Zero Energy Homes	{t₁, t₂, t₄, t₇}	65%	11%
Heat Exchanger Network Retrofit	{t₇, t₉, t₁₂}	79%	23%
Genetic Algorithms for Process Optimization	{t₁₀, t₁₂}	70%	25%
Progressive Crude Distillation	{t₉, t₁₁}	79%	34%
Water Management	{t₅, t₇, t₈}	65%	23%
Design of LNG Facilities	{t₃, t₄, t₅, t₆}	70%	23%

Table 5

Heights of all directed hyperedges

h (ξ₁)	(0.90, 0.11)	h (ξ₂)	(0.83, 0.23)
h (ξ₃)	(0.91, 0.20)	h (ξ₄)	(0.91, 0.23)
h (ξ₅)	(0.97, 0.03)	h (ξ₆)	(0.97, 0.03)
h (ξ₇)	(0.93, 0.12)

Table 6

Algorithm for powerful collaboration

Algorithm 4
1 . m =input(‘enter the number of trainees’);
2 . T =input(‘enter the degrees of membership of vertices(trainees) as m × 2’);
3 . r =input(‘enter the number of directed hyperedges’);
4 . Xi =input(‘enter the degrees of membership of directed hyperedges r × 2’);
5 . Y =input(‘enter the set valued function that tells us how many vertices are
contained in a hyperedge as r × m’);
6 . J=[zeros(r,1) ones(r,1)];
7 . fori = 1 : r
8 . fork = 1 : m
9 . ifY (i, k) = =1;
10 . J (i, 1) = max(J (i, 1) , T (k, 1));
11 . T (i, 2) = min(J (i, 2) , T (k, 2));
12 . end
13 . end
14 . end
15 . H = max(J (: , 1));j=0;v=zeros(r,2); b=1;
16 . forl = 1 : r
17 . if J(l,1)==H
18 . j=j+1;v(l,1)=l;b=min(b,J(l,2));
19 . end
20 . end
21 . if j>1
22 . forl = 1 : r
23 . if J(l,2)==b
24 . k=k+1;v(l,2)=l;
25 . fprintf(‘you can choice (any of these) hyperedge(s) % d’,l)
26 . end
27 . end
28 . else
29 . forl = 1 : r
30 . if J(l,1)==H
31 . fprintf(‘you can choice edge % d’,l)
32 . end
33 . end
34 . end

4.2.2 The enduring connection between projects:

Now, the line graph of the above 7-ROFDH can be used to determine the common trainees of distinct projects. The corresponding line graph is shown in Figure 13. The dashed lines between the projects demonstrate that they share some common trainees. The truth-membership and falsity-membership of these edges are given here. $\begin{matrix} (T_{p_{1} p_{2}}, F_{p_{1} p_{2}}) & = & (0.80, 0.11) \\ (T_{p_{1} p_{5}}, F_{p_{1} p_{5}}) & = & (0.80, 0.11) \\ (T_{p_{1} p_{6}}, F_{p_{1} p_{6}}) & = & (0.80, 0.11) \\ (T_{p_{2} p_{5}}, F_{p_{2} p_{5}}) & = & (0.79, 0.23) \\ (T_{p_{2} p_{3}}, F_{p_{2} p_{3}}) & = & (0.79, 0.23) \\ (T_{p_{3} p_{4}}, F_{p_{3} p_{4}}) & = & (0.79, 0.25) \end{matrix}$ The maximum truth-membership and minimum falsity-membership reveal the robust connection among the distinct projects. For instance, projects p₁ and p₅ are 80% connected to each other, i.e., the trainees of these projects can share their ideas, creative thinkings and motives among themselves to enhance the output of their projects. The method adopted in this section can be explained by a simple algorithm given in Table 7.

Fig. 13

Line graph of 7-ROFDH model to illustrate the common trainees.

Table 7

The enduring connection between projects

Algorithm 5
1 . m =input(‘enter the number of vertices’);
2 . T =input(‘enter the degrees of membership of vertices as m × 2’);
3 . r =input(‘enter the number of directed hyperedges’);
4 . Xi =input(‘enter the degrees of membership of directed hyperedges r × 2’);
5 . s =input(‘enter the number of edges in line graph’);
6 . P =input(‘enter the degrees of membership of edges s × 2’);
7 .H = max(P (: , 1));j=0;v=zeros(s,2); b=1;
8 . for n=1:s
9 . if P(n,1)==H
10 . j=j+1;v(n,1)=l;b=min(b,P(n,2));
11 . end
12 . end
13 . if j>1
14 . for n=1:s
15 . if P(n,2)==b
16 . k=k+1;v(n,2)=n;
17 . fprintf(‘you can choice (any of these) hyperedge(s) % d’,n)
18 . end
19 . end
20 . else
21 . for n=1:s
22 . if P(n,1)==H
23 . fprintf(‘you can choice edge % d’,n)
24 . end
25 . end
26 . end

5 Conclusions and future directions

A q-ROF model is an extension of fuzzy, IF and PF models. This model deals with real life applications more precisely and adequately. There are various phenomena, which cannot be well described through IF and PF models. In such cases, q-ROF model is more effective and powerful as it enlarges the space of vague information and relaxes the condition on the sum of truth-membership and falsity-membership degrees. In this research article, we have applied the concept of q-ROFS to directed hypergraphs. Directed hypergraphs, as generalization of digraphs, appear very useful in various fields of research. They are used to model functional systems in computer science, to check functional dependency in databases, in expert systems and routing protocols. A wider range of fuzzy information can be expressed through the methods proposed in this paper and they can handle the data for which the sum of truth and falsity degree is greater than one. So their scope of application is wider and closer to the real decision-making environment. Moreover, our proposed model generalizes the IF and PF directed hypergraphs. Although, IF and PF directed hypergraphs are also a good way to deal with truth-membership and falsity-membership, but q-ROFDHs are more generic and flexible as compared to these models. A q-ROFDH model not only widen the range of uncertain information and data but also reduces to IFDH model, when q = 1 and to PFDH model, when q = 2. Thus, it is more beneficial to consider a q-ROFDH model to handle truth-membership and falsity-membership degrees appearing in directed hypernetworks. Since, the space of q-ROFDHs depends on variable q, as the value of q is increased, the range of applicability of our proposed model also increases. Thus, the orthopairs of IFDHs and PFDHs can be handled through q-ROFDHs but the converse does not hold when q ≠ 1, 2. We have described q-ROFDHs and some of their properties. We have discussed the coloring of q-ROFDHs. Further, we have illustrated some interesting applications of q-ROFDHs in real life problems to explain the flexibility of the model when the given data possesses the uncertain behavior. We aim to broaden our study, (1) Interval valued bipolar neutrosophic hypergraphs, (2) Fuzzy rough soft directed hypergraphs, (3) Fuzzy rough neutrosophic hypergraphs, (4) Hesitant q-rung orthopair fuzzy hypergraphs and (5) q-Rung picture fuzzy hypergraphs.

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