Orientation-Uncertain HyperGraph and SuperHyperGraph Models for FuzzDirected, and Single-Valued NeutroDirected Graphs

1. Introduction

Many real-world phenomena cannot be adequately described by a purely binary notion of membership. In particular, vagueness, partial acceptance, hesitation, and incomplete information often require mathematical representations richer than ordinary crisp sets. A fundamental step in this direction was the introduction of fuzzy sets by Zadeh in 1965, where the membership of an element is quantified by a value in $[0,1]$ [1]. This idea was later extended by Atanassov through intuitionistic fuzzy sets [2,3], which distinguish membership from non-membership and retain a residual degree of hesitation. Another important development is the theory of neutrosophic sets [4,5], in which truth, indeterminacy, and falsity are represented by three components. Such a representation is particularly suitable for situations involving incomplete, uncertain, or potentially contradictory information.

These generalized uncertainty frameworks have naturally motivated corresponding developments in graph theory. Fuzzy graph models attach graded information to vertices, edges, or both [6,7]. Intuitionistic fuzzy graphs further distinguish positive and negative membership information [8], while neutrosophic graphs introduce an additional indeterminacy component into the network representation [9,10,11,12]. Other related constructions, such as plithogenic graphs, incorporate contradiction-based mechanisms among attributes [13]. Similar ideas have also been developed in the broader settings of hypergraphs and SuperHyperGraphs [7,14].

A common feature of many such models is that uncertainty is imposed on the vertices, edges, incidences, or associated degree functions, whereas the directional structure of an edge is usually prescribed in advance. This suggests a different modeling problem. Instead of assuming that the orientation is fixed, one may ask whether the orientation itself can carry uncertainty. For instance, an interaction between two vertices u and v may support the direction $u\to v$ to one degree and the reverse direction $v\to u$ to another degree, while an undirected or bidirected interpretation may also remain admissible.

This perspective leads to orientation-based uncertainty models such as the Fuzzy-Oriented Mixed Graph and the Neutrosophic-Oriented Mixed Graph. In these constructions, the uncertain information is attached directly to possible orientation types rather than being restricted to the existence or membership of vertices and edges. Consequently, they offer a convenient framework for representing network interactions whose direction cannot be described satisfactorily by a single deterministic orientation.

The aim of this paper is to transfer this orientation-uncertainty viewpoint from ordinary graphs to higher-order combinatorial structures. To this end, we introduce the FuzzDirected HyperGraph, the Single-Valued NeutroDirected HyperGraph, the FuzzDirected n-SuperHyperGraph, and the Single-Valued NeutroDirected n-SuperHyperGraph. In these models, fuzzy or single-valued neutrosophic information is assigned to admissible orientations of hyperedges and superhyperedges. The resulting constructions extend orientation-based uncertainty modeling beyond pairwise relations and provide a foundation for studying uncertain directional behavior in higher-order and hierarchical network systems.

2. Preliminaries

We begin by reviewing the basic terminology and notation used throughout this paper.

2.1. Fuzzy and Neutrosophic Set

A fuzzy set assigns each element a membership degree in $[0,1]$, modeling gradual belonging instead of binary inclusion [1,15]. A neutrosophic set assigns truth, indeterminacy, and falsity degrees to each element, allowing incomplete and inconsistent information simultaneously [4,16].

Definition 1

(Fuzzy set). [1] Let Y be a non-empty universe. A fuzzy set τ on Y is a function 

$$\tau :Y⟶[0,1],$$

 assigning to each $y\in Y$ a membership value $\tau \left(y\right)$. A fuzzy relation on Y is a fuzzy subset δ of $Y\times Y$. Given a fuzzy set τ on Y, the relation δ is said to be a fuzzy relation on τ whenever 

$$\delta (y,z)\le min\left\{\tau \right(y),\tau (z\left)\right\},\forall y,z\in Y.$$

Definition 2

(Neutrosophic set). [5] Let X be a nonempty set. A neutrosophic set A on X is specified by three functions 

$$T_A:X\to [0,1],I_A:X\to [0,1],F_A:X\to [0,1],$$

 where, for each $x\in X$, the numbers $T_A\left(x\right)$, $I_A\left(x\right)$, and $F_A\left(x\right)$ represent, respectively, the degrees of truth-membership, indeterminacy, and falsity-membership of x in A. They satisfy the admissibility constraint 

$$0\le T_A\left(x\right)+I_A\left(x\right)+F_A\left(x\right)\le 3(x\in X).$$

2.2. Probabilistic Mixed-Orientation Graph

A probabilistic mixed-orientation graph assigns to each edge a probability distribution over the possible orientation types in $\mathcal{D}$. Thus, the uncertainty concerns the orientation type of an existing edge rather than the existence of the edge itself.

Definition 3

(Probabilistic mixed-orientation graph). A probabilistic mixed-orientation graph is a triple 

$$\mathcal{G}=(V,E,{\left\{p_e\right\}}_{e\in E}),$$

 where, for each $e\in E$, 

$$p_e:\mathcal{D}⟶[0,1]$$

 is a probability mass function satisfying 

$$\sum _{d\in \mathcal{D}}{p}_e\left(d\right)=1.$$

Unless otherwise stated, the orientation types of distinct edges are sampled independently. Hence a realization is determined by mutually independent random variables 

$$D\left(e\right):\Omega ⟶\mathcal{D},e\in E,$$

 such that 

$$Pr(D\left(e\right)=d)=p_e\left(d\right)(d\in \mathcal{D}).$$

Remark 1.

The independence assumption is not mathematically necessary for a more general theory. One may instead prescribe an arbitrary joint probability distribution on 

$${\mathcal{D}}^E.$$

The present definition corresponds to the canonical product model determined by the edge-wise marginal distributions $p_e$. 

2.3. Fuzzdirected Graph (Fuzzy-Oriented Mixed Graph)

A Fuzzdirected Graph, also called a fuzzy-oriented mixed graph, assigns fuzzy membership degrees to the possible orientation types of each edge. Thus, unlike a conventional fuzzy digraph, the primary source of fuzziness is the orientation type itself.

Definition 4

(Fuzzdirected Graph). A Fuzzdirected Graph is a triple 

$$\tilde{\mathcal{G}}=(V,E,{\left\{{\mu }_e\right\}}_{e\in E}),$$

 where 

$${\mu }_e:\mathcal{D}⟶[0,1]$$

 is a fuzzy set on $\mathcal{D}$ for every $e\in E$. 

For 

$$\alpha \in (0,1],$$

 define the α-cut 

$$D_{\alpha }\left(e\right):=\left\{d\in \mathcal{D}\mid {\mu }_e\left(d\right)\ge \alpha \right\}.$$

The Fuzzdirected Graph is called normal if 

$$\underset{d\in \mathcal{D}}{max}{\mu }_e\left(d\right)=1$$

 for every $e\in E$. 

2.4. Single-Valued NeutroDirected Graph (Neutrosophic-Oriented Mixed Graph)

A single-valued NeutroDirected Graph assigns a single-valued neutrosophic triple to each possible orientation type of every underlying edge.

Definition 5

(Single-valued NeutroDirected Graph). A single-valued NeutroDirected Graph is a triple 

$$\tilde{\mathcal{G}}=\left(V,E,{\left\{(T_e,I_e,F_e)\right\}}_{e\in E}\right),$$

 where 

$$T_e,I_e,F_e:\mathcal{D}⟶[0,1]$$

 for every $e\in E$. 

For each $d\in \mathcal{D}$, the triple 

$$\left(T_e\left(d\right),I_e\left(d\right),F_e\left(d\right)\right)\in {[0,1]}^3$$

 represents, respectively, the truth-membership, indeterminacy-membership, and falsity-membership associated with the orientation type d. 

Since all three components belong to $[0,1]$, one automatically has 

$$0\le T_e\left(d\right)+I_e\left(d\right)+F_e\left(d\right)\le 3.$$

2.5. SuperHyperGraphs

A hypergraph generalizes a graph by allowing each hyperedge to connect any nonempty subset of vertices, representing higher-order relationships directly. A SuperHyperGraph extends hypergraphs by allowing vertices to be set-valued objects from iterated powersets, with superedges connecting such structured vertices. We next recall the basic definitions for undirected SuperHyperGraphs.

Definition 6

(Iterated powersets). Let S be a set and let $n\in {\mathbb{N}}_0$. Define the iterated powersets of S recursively by 

$${\mathcal{P}}^0\left(S\right)S,{\mathcal{P}}^{n+1}\left(S\right)\mathcal{P}\left({\mathcal{P}}^n\left(S\right)\right)(n\ge 0).$$

When convenient, we also use the nonempty powerset

$${\mathcal{P}}_{*}\left(S\right)\mathcal{P}\left(S\right)\setminus \{\varnothing \},$$

 and the corresponding iterated nonempty powersets, defined by ${\mathcal{P}}_{*}^0\left(S\right)S$ and 

$${\mathcal{P}}_{*}^{n+1}\left(S\right){\mathcal{P}}_{*}\left({\mathcal{P}}_{*}^n\left(S\right)\right)(n\ge 0).$$

Definition 7

(Hypergraph). [17,18] A hypergraph is a pair $H=\left(V\right(H),E(H\left)\right)$ consisting of 

• a (typically finite) vertex set $V\left(H\right)$, and 
• a family of nonempty subsets $E\left(H\right)\subseteq \mathcal{P}\left(V\right(H\left)\right)\setminus \{\varnothing \}$, whose elements are called hyperedges.

If every hyperedge has cardinality r, then H is calledr-uniform ; in particular, 2-uniform hypergraphs coincide with simple graphs. 

Definition 8 (2-section (shadow graph)). Let $H=(V,E)$ be a hypergraph. The 2-section (or shadow graph ) of H is the simple graph ${\left[H\right]}_2=(V,E_2)$ defined by 

$$E_2\left\{\{u,v\}\subseteq V|u\ne v,\exists e\in E\mathrm{with}\{u,v\}\subseteq e\right\}.$$

Definition 9

(n-SuperHyperGraph). [14,19,20] Let $V_0$ be a nonempty (typically finite) base set and fix $n\in {\mathbb{N}}_0$. Ann-SuperHyperGraph on $V_0$ is a hypergraph 

$${\mathrm{SuHyG}}^{\left(n\right)}=(V,\mathcal{E})$$

 such that 

$$V\subseteq {\mathcal{P}}^n\left(V_0\right),\mathcal{E}\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}.$$

Elements of V are called n-supervertices, and elements of $\mathcal{E}$ are called n-superedges. Thus, a superedge is a nonempty set of n-supervertices, while each n-supervertex may itself be a set-valued object living in the iterated powerset hierarchy ${\mathcal{P}}^n\left(V_0\right)$. 

3. Main Results

In this section, we present the main results of this paper. For every nonempty set e, define the family of ordered bipartitions

$$Bip\left(e\right):=\left\{(A,B)\in {\mathcal{P}}_{*}\left(e\right)\times {\mathcal{P}}_{*}\left(e\right)|A\cap B=\varnothing ,A\cup B=e\right\},$$

and the corresponding family of unordered bipartitions

$$UBip\left(e\right):=\left\{\{A,B\}\left|\right(A,B)\in Bip(e)\right\}.$$

The admissible orientation space of e is

$$\mathcal{D}\left(e\right):=\left\{{\mathsf{U}}_e\right\}\cup \left\{(\mathsf{D},A,B)\left|\right(A,B)\in Bip(e)\right\}\cup \left\{(\mathsf{B},\{A,B\left\}\right)\left|\right\{A,B\}\in UBip(e)\right\}.$$

Here, ${\mathsf{U}}_e$ denotes the undirected orientation, $(\mathsf{D},A,B)$ denotes the orientation from the tail set A to the head set B, and $(\mathsf{B},\{A,B\left\}\right)$ denotes the bidirected orientation between the two blocks A and B. In particular, if $e=\{u,v\}$, then

$$\mathcal{D}\left(e\right)=\left\{{\mathsf{U}}_e,(\mathsf{D},\left\{u\right\},\left\{v\right\}),(\mathsf{D},\left\{v\right\},\left\{u\right\}),(\mathsf{B},\{\left\{u\right\},\left\{v\right\}\})\right\},$$

which gives exactly the usual four mixed-orientation types.

3.1. FuzzDirected HyperGraph

A FuzzDirected HyperGraph extends a FuzzDirected Graph by assigning fuzzy degrees to the admissible orientations of each hyperedge.

Definition 10

(FuzzDirected HyperGraph). Let 

$$H=(V,E),E\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \},$$

 be a hypergraph. A FuzzDirected HyperGraph is a structure 

$${\tilde{H}}_F=\left(V,E,{\left\{{\mu }_e\right\}}_{e\in E}\right),$$

 where, for every $e\in E$, 

$${\mu }_e:\mathcal{D}\left(e\right)⟶[0,1].$$

For $d\in \mathcal{D}\left(e\right)$, the value ${\mu }_e\left(d\right)$ represents the fuzzy degree to which the hyperedge e has orientation d. No normalization condition such as ${\sum }_{d\in \mathcal{D}\left(e\right)}{\mu }_e\left(d\right)=1$ is required. 

Theorem 1

(Well-definedness of a FuzzDirected HyperGraph). For every hypergraph $H=(V,E)$, the FuzzDirected HyperGraph 

$${\tilde{H}}_F=\left(V,E,{\left\{{\mu }_e\right\}}_{e\in E}\right)$$

 defined above is well-defined. Moreover, on a 2-uniform hypergraph, the definition reduces to the corresponding FuzzDirected Graph model. 

Proof. 

For every $e\in E$, one has $e\ne \varnothing$. Hence $Bip\left(e\right)$, $UBip\left(e\right)$, and therefore $\mathcal{D}\left(e\right)$ are well-defined sets. Moreover, ${\mathsf{U}}_e\in \mathcal{D}\left(e\right)$, so

$$\mathcal{D}\left(e\right)\ne \varnothing .$$

Thus every map

$${\mu }_e:\mathcal{D}\left(e\right)\to [0,1]$$

is an ordinary fuzzy set on a nonempty orientation space. Consequently, the family ${\left\{{\mu }_e\right\}}_{e\in E}$ is well-defined.

If H is 2-uniform, then every edge $e=\{u,v\}$ has exactly the four orientation types displayed above: undirected, $u\to v$, $v\to u$, and bidirected. Hence the construction specializes to the FuzzDirected Graph case. □

3.2. Single-Valued NeutroDirected HyperGraph

A Single-Valued NeutroDirected HyperGraph associates truth, indeterminacy, and falsity degrees with every admissible orientation of each hyperedge.

Definition 11

(Single-Valued NeutroDirected HyperGraph). Let 

$$H=(V,E),E\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \},$$

 be a hypergraph. A Single-Valued NeutroDirected HyperGraph is a structure 

$${\tilde{H}}_N=\left(V,E,{\left\{(T_e,I_e,F_e)\right\}}_{e\in E}\right),$$

 where, for every $e\in E$, 

$$T_e,I_e,F_e:\mathcal{D}\left(e\right)⟶[0,1].$$

For every $d\in \mathcal{D}\left(e\right)$, 

$$\left(T_e\left(d\right),I_e\left(d\right),F_e\left(d\right)\right)\in {[0,1]}^3$$

 represents respectively the truth, indeterminacy, and falsity degrees associated with orientation d. 

Theorem 2

(Well-definedness of a Single-Valued NeutroDirected HyperGraph). For every hypergraph $H=(V,E)$, the Single-Valued NeutroDirected HyperGraph defined above is well-defined. In particular, 

$$0\le T_e\left(d\right)+I_e\left(d\right)+F_e\left(d\right)\le 3$$

 for every $e\in E$ and $d\in \mathcal{D}\left(e\right)$. For a 2-uniform hypergraph, the construction reduces to the Single-Valued NeutroDirected Graph model. 

Proof. 

Since every $e\in E$ is nonempty, the orientation space $\mathcal{D}\left(e\right)$ is nonempty and well-defined. Therefore the three functions

$$T_e,I_e,F_e:\mathcal{D}\left(e\right)\to [0,1]$$

have well-defined common domain and codomain.

For every $d\in \mathcal{D}\left(e\right)$,

$$0\le T_e\left(d\right),I_e\left(d\right),F_e\left(d\right)\le 1,$$

and hence

$$0\le T_e\left(d\right)+I_e\left(d\right)+F_e\left(d\right)\le 3.$$

Thus each orientation is assigned a valid single-valued neutrosophic triple. When $\left|e\right|=2$, $\mathcal{D}\left(e\right)$ consists of the four ordinary mixed-orientation types, giving the corresponding graph model. □

3.3. FuzzDirected n-SuperHyperGraph

The FuzzDirected construction extends naturally to an n-SuperHyperGraph because each superedge is itself a nonempty set of n-supervertices.

Definition 12

(FuzzDirected n-SuperHyperGraph). Let $V_0\ne \varnothing$, let $n\in {\mathbb{N}}_0$, and let 

$${\mathrm{SuHyG}}^{\left(n\right)}=(V,\mathcal{E})$$

 be an n-SuperHyperGraph satisfying 

$$V\subseteq {\mathcal{P}}^n\left(V_0\right),\mathcal{E}\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}.$$

A FuzzDirected n-SuperHyperGraph is a structure 

$${\widetilde{\mathrm{SuHyG}}}_F^{\left(n\right)}=\left(V_0,V,\mathcal{E},{\left\{{\mu }_S\right\}}_{S\in \mathcal{E}}\right),$$

 where, for every n-superedge $S\in \mathcal{E}$, 

$${\mu }_S:\mathcal{D}\left(S\right)⟶[0,1].$$

For $d\in \mathcal{D}\left(S\right)$, the value ${\mu }_S\left(d\right)$ represents the fuzzy degree to which the superedge S has orientation d. 

Theorem 3

(Well-definedness of a FuzzDirected n-SuperHyperGraph). Every FuzzDirected n-SuperHyperGraph defined above is well-defined. Furthermore, for $n=0$, the construction reduces to a FuzzDirected HyperGraph. 

Proof. 

Since

$$\mathcal{E}\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \},$$

every $S\in \mathcal{E}$ is a nonempty set of n-supervertices. Hence

$$Bip\left(S\right)\subseteq {\mathcal{P}}_{*}\left(S\right)\times {\mathcal{P}}_{*}\left(S\right)$$

and $\mathcal{D}\left(S\right)$ are well-defined. In particular,

$${\mathsf{U}}_S\in \mathcal{D}\left(S\right),$$

so $\mathcal{D}\left(S\right)\ne \varnothing$, and therefore

$${\mu }_S:\mathcal{D}\left(S\right)\to [0,1]$$

is well-defined.

Importantly, the orientation is imposed on the elements of the superedge S, namely the n-supervertices, and does not depend on their internal representation in ${\mathcal{P}}^n\left(V_0\right)$. Hence no type inconsistency is introduced.

For $n=0$,

$${\mathcal{P}}^0\left(V_0\right)=V_0,$$

so $V\subseteq V_0$, and the underlying 0-SuperHyperGraph is an ordinary hypergraph. Therefore the definition reduces to the FuzzDirected HyperGraph construction. □

3.4. Single-Valued NeutroDirected n-SuperHyperGraph

The neutrosophic orientation model can likewise be defined on superedges by assigning a single-valued neutrosophic triple to each admissible orientation.

Definition 13

(Single-Valued NeutroDirected n-SuperHyperGraph). Let $V_0\ne \varnothing$, let $n\in {\mathbb{N}}_0$, and let 

$${\mathrm{SuHyG}}^{\left(n\right)}=(V,\mathcal{E})$$

 be an n-SuperHyperGraph satisfying 

$$V\subseteq {\mathcal{P}}^n\left(V_0\right),\mathcal{E}\subseteq \mathcal{P}\left(V\right)\setminus \{\varnothing \}.$$

A Single-Valued NeutroDirected n-SuperHyperGraph is a structure 

$${\widetilde{\mathrm{SuHyG}}}_N^{\left(n\right)}=\left(V_0,V,\mathcal{E},{\left\{(T_S,I_S,F_S)\right\}}_{S\in \mathcal{E}}\right),$$

 where, for every $S\in \mathcal{E}$, 

$$T_S,I_S,F_S:\mathcal{D}\left(S\right)⟶[0,1].$$

For each $d\in \mathcal{D}\left(S\right)$, 

$$\left(T_S\left(d\right),I_S\left(d\right),F_S\left(d\right)\right)$$

 is the single-valued neutrosophic degree associated with orientation d of the superedge S. 

Theorem 4

(Well-definedness of a Single-Valued NeutroDirected n-SuperHyperGraph). Every Single-Valued NeutroDirected n-SuperHyperGraph defined above is well-defined. For every $S\in \mathcal{E}$ and $d\in \mathcal{D}\left(S\right)$, 

$$0\le T_S\left(d\right)+I_S\left(d\right)+F_S\left(d\right)\le 3.$$

Moreover, for $n=0$, the construction reduces to a Single-Valued NeutroDirected HyperGraph. 

Proof. 

Every $S\in \mathcal{E}$ is a nonempty subset of V, and every element of S is an n-supervertex belonging to ${\mathcal{P}}^n\left(V_0\right)$. Therefore $Bip\left(S\right)$, $UBip\left(S\right)$, and $\mathcal{D}\left(S\right)$ are well-defined, with

$${\mathsf{U}}_S\in \mathcal{D}\left(S\right).$$

Hence the maps

$$T_S,I_S,F_S:\mathcal{D}\left(S\right)\to [0,1]$$

are well-defined.

Since every component belongs to $[0,1]$,

$$0\le T_S\left(d\right)+I_S\left(d\right)+F_S\left(d\right)\le 3$$

for every $d\in \mathcal{D}\left(S\right)$. The construction therefore assigns a valid single-valued neutrosophic triple to every admissible orientation of every superedge.

Finally, if $n=0$, then ${\mathcal{P}}^0\left(V_0\right)=V_0$, so the underlying structure is an ordinary hypergraph. The definition consequently reduces to the Single-Valued NeutroDirected HyperGraph case. □

4. Conclusions

In this paper, we extended orientation-uncertainty frameworks from ordinary graphs to higher-order network structures based on HyperGraphs and SuperHyperGraphs. In particular, we introduced fuzzy- and single-valued neutrosophic-oriented extensions in which uncertainty is associated with the possible orientations of hyperedges and superhyperedges.

As directions for future research, we expect further extensions based on other uncertainty frameworks, including Plithogenic Sets [21,22], Double-Valued Neutrosophic Sets [23,24], and HyperFuzzy Sets [25,26]. It would also be worthwhile to investigate graph algorithms, computational methods, and computational complexity for the proposed orientation-uncertain HyperGraph and SuperHyperGraph models. Such studies may further develop both the theoretical foundations and computational aspects of orientation uncertainty in higher-order network structures.