**1. Introduction**

In mathematical modeling, competition graphs are sufficient to specify well defined behaviors of objects and specifically predator–prey relations. In 1968, while studying applications of graph theory in ecology, Cohen introduced the notion of a competition graph. Competition graphs have been applied to various fields of biological sciences and technology. After the strong motivation of energy and food competition in food webs between species, competition graphs were a part of active research in recent years. In 2004, Sonntag and Teichert [1] introduced the notion of competition hypergraphs. These representations are crisp hypergraphs that do not describe all the competitions of real-world problems. These models contain uncertainty and fuzzy in nature for problems that are more relevant to everyday life, including critical writing style of a writer, predator–prey relationship, trading relationship among different communities, honesty leadership quality of a politician and, signal strength of wireless devices. Motivating from this idea, we have applied the notion of fuzzy sets to competition hypergraphs to study the problems having nonlinear uncertainties.

In 1965, Zadeh [2] introduced the strong mathematical notion of fuzzy set in order to discuss the phenomena of vagueness and uncertainty in various real-life problems. Using the concept of fuzzy relations introduced by Zadeh [3], the idea of fuzzy graph was given by Kaufmann [4]. The fuzzy relations in fuzzy sets were studied by Rosenfeld [5] and he introduced the structure of fuzzy graphs, obtaining analysis of various graph theoretical concepts. Lee-kwang and Lee [6] redefined and extended the notion of fuzzy hypergraphs whose idea was first discussed by Kaufmann [4]. Later, the idea of fuzzy hypergraph was studied by Goetschel in [7,8]. The concept of interval-valued fuzzy hypergraphs was initiated by Chen [9] and Parvathi et al. [10] generalized the idea of hypergraphs to intuitionistic fuzzy hypergraphs. Moreover, Akram and Dudek [11], Akram and Luqman [12–14], and Akram and Shahzadi [15] have discussed certain extensions of fuzzy hypergraphs with applications.

Samanta and Pal [16] studied fuzzy *k*-competition graphs and *p*-competition graphs. Later, Samanta et al. [17] introduced the concept of *m*-step fuzzy competition graphs. Applying the idea of bipolar fuzzy sets to competition graphs, Alshehri and Akram [18] introduced the notion of bipolar fuzzy competition graphs and applied this idea to economic systems. Furthermore, the study of bipolar fuzzy competition graphs was discussed by Sarwar and Akram in [19]. Certain competition graphs based on neutrosophic environment were described in [20,21]. In this research paper, we introduce the concept of fuzzy competition hypergraphs as a generalized case of fuzzy competition graphs. We study various new concepts, including fuzzy column hypergraphs, fuzzy row hypergraphs, fuzzy competition hypergraphs, fuzzy *k*-competition hypergraphs and fuzzy neighbourhood hypergraphs and investigate some of their interesting properties. We design certain algorithms for the construction of different types of fuzzy competition hypergraphs. We also present applications of fuzzy competition hypergraphs in decision support systems, including food webs, social networks and business marketing.

We have used basic notions and terminologies in this research paper. For other terminologies, notations and definitions not given in the paper, the readers are referred to [2,3,5,9,10,17,19,22–36].

**Definition 1.** *A fuzzy hypergraph on a non-empty set X is a pair H* = (*μ*, *ρ*) *where μ* = {*μ*1, *μ*2, ... , *μr*}*, μi* : *X* → [0, 1] *are fuzzy subsets on X such that* A *i supp*(*μi*) = *X, for all μi* ∈ *μ. ρ is a fuzzy relation on the fuzzy subsets μi such that*

*ρ*(*Ei*) ≤ min{*μi*(*<sup>x</sup>*1), *μi*(*<sup>x</sup>*2),..., *μi*(*xs*)}, *Ei* = {*<sup>x</sup>*1, *x*2,..., *xs*}, *for all x*1, *x*2,..., *xs* ∈ *X*.

#### **2. Fuzzy Competition Hypergraphs**

In this section, we discuss various types of fuzzy competition hypergraphs with certain properties and algorithms.

**Definition 2.** *Let A* = [*xij*]*n*×*n be the adjacency matrix of a fuzzy digraph* #» *G* = (*μ*, #» *λ*) *on a non-empty set X. The fuzzy row hypergraph of* #» *G, denoted by* R◦H(#» *<sup>G</sup>*)=(*μ*, *<sup>λ</sup>r*)*, having the same set of vertices as* #» *G and the set of hyperedges is defined as*

$$\left\{ \left\{ \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_r \right\} \middle| A(\mathbf{x}\_{ij}) > 0, r \ge 2, \text{ for each } 1 \le i \le r, \mathbf{x}\_i \in X, \text{ for some } 1 \le j \le n \right\}.$$

*The degree of membership of hyperedges is defined as*

$$\lambda\_I(\{\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_I\}) = \left[\mu(\mathbf{x}\_1) \land \mu(\mathbf{x}\_2) \land \dots \land \mu(\mathbf{x}\_\delta)\right] \times \max\_j \{\overrightarrow{\lambda}(\mathbf{x}\_1 \mathbf{x}\_j) \land \overleftrightarrow{\lambda}(\mathbf{x}\_2 \mathbf{x}\_j) \land \dots \land \overleftrightarrow{\lambda}(\mathbf{x}\_\delta \mathbf{x}\_j)\}.$$

**Definition 3.** *The fuzzy column hypergraph of* #» *G, denoted by* C◦H(#» *<sup>G</sup>*)=(*μ*, *<sup>λ</sup>cl*)*, having the same set of vertices as* #» *G and the set of hyperedges is defined as*

$$\left\{ \left\{ \mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_s \right\} \middle| A(\mathbf{x}\_{ji}) > 0, s \ge 2, \text{ for each } 1 \le i \le s, \mathbf{x}\_i \in X, \text{ for some } 1 \le j \le n \right\}.$$

*The degree of membership of hyperedges is defined as*

$$\lambda\_{\mathrm{cl}}(\{\mathbf{x}\_1, \mathbf{x}\_2, \dots, \mathbf{x}\_s\}) = \left[\mu(\mathbf{x}\_1) \land \mu(\mathbf{x}\_2) \land \dots \land \mu(\mathbf{x}\_s)\right] \times \max\_j \{ \overrightarrow{\lambda}(\mathbf{x}\_j \mathbf{x}\_1) \land \overleftarrow{\lambda}(\mathbf{x}\_j \mathbf{x}\_2) \land \dots \land \overleftarrow{\lambda}(\mathbf{x}\_j \mathbf{x}\_s) \}.$$

The methods for computing fuzzy row hypergraph and fuzzy column hypergraph are given in Algorithms A1 and A2, respectively.

**Example 1.** *Consider the universe X* = {*<sup>x</sup>*1, *x*2, *x*3, *x*4, *x*5, *<sup>x</sup>*6}*, μ a fuzzy set on X and* #» *λ a fuzzy relation in X as defined in Tables 1 and 2, respectively. The fuzzy digraph* #» *G* = (*μ*, #» *λ*) *is shown in Figure 1. The adjacency matrix of* #» *G is given in Table 3.*

*Using Algorithm A1 and Table 3, there are three hyperedges E*2 = {*<sup>x</sup>*1, *x*5, *<sup>x</sup>*6}*, E*3 = {*<sup>x</sup>*2, *<sup>x</sup>*5} *and E*4 = {*<sup>x</sup>*3, *<sup>x</sup>*5}*, corresponding to the columns x*2, *x*3 *and x*4 *of adjacency matrix, in fuzzy row hypergraph of* #»*G. The membership degree of the hyperedges is calculated as*

*<sup>λ</sup>r*(*<sup>E</sup>*2) = I*μ*(*<sup>x</sup>*1) ∧ *μ*(*<sup>x</sup>*5) ∧ *<sup>μ</sup>*(*<sup>x</sup>*6)J × I*<sup>x</sup>*12 ∧ *x*52 ∧ *<sup>x</sup>*62J = 0.3 × 0.3 = 0.09*,*

*<sup>λ</sup>r*(*<sup>E</sup>*3) = I*μ*(*<sup>x</sup>*2) ∧ *<sup>μ</sup>*(*<sup>x</sup>*5)J × I*<sup>x</sup>*23 ∧ *<sup>x</sup>*53J = 0.4 × 0.1 = 0.04*,*

*<sup>λ</sup>r*(*<sup>E</sup>*4) = I*μ*(*<sup>x</sup>*3) ∧ *<sup>μ</sup>*(*<sup>x</sup>*5)J × I*<sup>x</sup>*34 ∧ *<sup>x</sup>*54J = 0.4 × 0.4 = 0.16*.*

*The fuzzy row hypergraph is shown in Figure 2. Using Algorithm A2 and Table 3, the hyperedges in fuzzy column hypergraph of* #» *G are E*1 = {*<sup>x</sup>*2, *<sup>x</sup>*6}*, E*5 = {*<sup>x</sup>*2, *x*3, *<sup>x</sup>*4} *and E*6 = {*<sup>x</sup>*2, *<sup>x</sup>*5}*, corresponding to the rows x*2, *x*5 *and x*6 *of the adjacency matrix. The membership degree of the hyperedges is calculated as*

*<sup>λ</sup>cl*(*<sup>E</sup>*5) = I*μ*(*<sup>x</sup>*2) ∧ *μ*(*<sup>x</sup>*3) ∧ *<sup>μ</sup>*(*<sup>x</sup>*4)J × I*<sup>x</sup>*52 ∧ *x*53 ∧ *<sup>x</sup>*54J = 0.4 × 0.3 = 0.12*,* 

*<sup>λ</sup>cl*(*<sup>E</sup>*1) = I*μ*(*<sup>x</sup>*2) ∧ *<sup>μ</sup>*(*<sup>x</sup>*6)J× I*<sup>x</sup>*12 ∧ *<sup>x</sup>*16J= 0.3 × 0.2 = 0.06*,* 

*<sup>λ</sup>cl*(*<sup>E</sup>*6) = I*μ*(*<sup>x</sup>*2) ∧ *<sup>μ</sup>*(*<sup>x</sup>*5)J× I*<sup>x</sup>*62 ∧ *<sup>x</sup>*65J= 0.4 × 0.1 = 0.04*.*

*The fuzzy column hypergraph is given in Figure 3.*

**Table 1.** Fuzzy vertex set *μ*.


**Figure 1.** Fuzzy digraph #» *G* .

**Figure 2.** R◦H(#»*G*).

**Figure 3.** C◦H(#»*G*).

**Table 2.** Fuzzy relation #» *λ*.


**Table 3.** Adjacency matrix.


**Definition 4.** *[25] A fuzzy digraph on a non-empty set X is a pair* #» *G* = (*μ*, #» *λ*) *of functions μ* : *X* → [0, 1] *and* #» *λ* : *X* × *X* → [0, 1]*, such that for all x*, *y* ∈ *X,* #» *<sup>λ</sup>*(*xy*) ≤ min{*μ*(*x*), *μ*(*y*)}*.*

**Definition 5.** *[16] A fuzzy out neighbourhood of a vertex x of a fuzzy digraph* #» *G* = (*μ*, #» *λ*) *is a fuzzy set* N +(*x*)=(*X*+*x* , *<sup>μ</sup>*<sup>+</sup>*x* ), *where <sup>X</sup>*+*x* = {*y*|#»*λ*(*xy*) > 0} *and <sup>μ</sup>*<sup>+</sup>*x* : *<sup>X</sup>*+*x* → [0, 1] *is defined by <sup>μ</sup>*<sup>+</sup>*x* (*y*) = #»*<sup>λ</sup>*(*xy*)*.*

**Definition 6.** *[16] The fuzzy in neighbourhood of vertex x of a fuzzy digraph is a fuzzy set* N <sup>−</sup>(*x*)=(*<sup>X</sup>*<sup>−</sup>*x* , *μ*<sup>−</sup>*x* ), *where X*<sup>−</sup>*x* = {*y*| #» *<sup>λ</sup>*(*yx*) > 0} *and μ*<sup>−</sup>*x* : *X*<sup>−</sup>*x* → [0, 1] *is defined by μ*<sup>−</sup>*x* (*y*) = #» *<sup>λ</sup>*(*yx*)*.*

**Definition 7.** *Let* #» *G* = (*μ*, #» *λ*) *be a fuzzy digraph on a non-empty set X. The fuzzy competition hypergraph* CH(#»*G*)=(*μ*, *<sup>λ</sup>c*) *on X having the same vertex set as* #»*G and there is a hyperedge consisting of vertices x*1, *x*2, ... , *xs if* N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xs*) = ∅*. The degree of membership of hyperedge E* = {*<sup>x</sup>*1, *x*2,..., *xs*} *is defined as*

$$\lambda\_{\mathbf{c}}(E) = [\mu(\mathbf{x}\_1) \land \mu(\mathbf{x}\_2) \land \dots \land \mu(\mathbf{x}\_s)] \times h(\mathcal{N}^+(\mathbf{x}\_1) \cap \mathcal{N}^+(\mathbf{x}\_2) \cap \dots \cap \mathcal{N}^+(\mathbf{x}\_s)),$$

*where h*(N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xs*)) *denotes the height of fuzzy set* N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xs*)*.*

The method for constructing fuzzy competition hypergraph of a fuzzy digraph is given in Algorithm A3.

**Table 4.** Fuzzy out

**Lemma 1.** *The fuzzy competition hypergraph of a fuzzy digraph* #» *G is a fuzzy row hypergraph of* #» *G.*

**Proof.** Let #» *G* = (*μ*, #» *λ*) be a fuzzy digraph; then, for any hyperedge *E* = {*<sup>x</sup>*1, *x*2,..., *xs*} of CH(#»*G*),

$$\begin{split} \lambda\_{\mathsf{c}}(E) &= \left[ \mu(\mathbf{x}\_{1}) \wedge \mu(\mathbf{x}\_{2}) \wedge \dots \wedge \mu(\mathbf{x}\_{s}) \right] \times h(\mathcal{N}^{+}(\mathbf{x}\_{1}) \cap \mathcal{N}^{+}(\mathbf{x}\_{2}) \cap \dots \cap \mathcal{N}^{+}(\mathbf{x}\_{s})) \\ &= \left[ \mu(\mathbf{x}\_{1}) \wedge \mu(\mathbf{x}\_{2}) \wedge \dots \wedge \mu(\mathbf{x}\_{s}) \right] \times \max\_{\stackrel{\scriptstyle}{j}} \{ \mathcal{N}^{+}(\mathbf{x}\_{1}) \cap \mathcal{N}^{+}(\mathbf{x}\_{2}) \cap \dots \cap \mathcal{N}^{+}(\mathbf{x}\_{s}) \} \\ &= \left[ \mu(\mathbf{x}\_{1}) \wedge \mu(\mathbf{x}\_{2}) \wedge \dots \wedge \mu(\mathbf{x}\_{s}) \right] \times \max\_{\stackrel{\scriptstyle}{j}} \{ \overline{\lambda}(\mathbf{x}\_{1} \mathbf{x}\_{j}) \wedge \overline{\lambda}(\mathbf{x}\_{2} \mathbf{x}\_{j}) \wedge \dots \wedge \overline{\lambda}(\mathbf{x}\_{\mathbb{R}} \mathbf{x}\_{j}) \} = \lambda\_{\mathsf{r}}(E) . \end{split}$$

It follows that *E* is a hyperedge of fuzzy row hypergraph.

**Example 2.** *Consider the fuzzy digraph given in Figure 1. The fuzzy out neighbourhood and fuzzy in neighbourhood of all the vertices are given in Table 4.*

*Using Algorithm A3, the relation f* : *X* → *X of* #» *G is given in Figure 4. The construction of fuzzy competition hypergraph from* #» *G is given as follows:*


*The fuzzy competition hypergraph is given in Figure 5. From Figures 2 and 5, it is clear that fuzzy competition hypergraph is a fuzzy row hypergraph.*

neighbouhood

 of vertices in #»

*G*.

 and fuzzy in

neighbourhood


**Figure 4.** Representation of fuzzy relation in #» *G*.

**Figure 5.** Fuzzy competition hypergraph CH(#»*G*).

**Definition 8.** *The fuzzy double competition hypergraph* DCH(#»*G*)=(*μ*, *<sup>λ</sup>d*) *having same vertex set as* #»*G and there is a hyperedge consisting of vertices x*1, *x*2, ... , *xs if* N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xs*) = ∅ *and* N <sup>−</sup>(*<sup>x</sup>*1) ∩ N <sup>−</sup>(*<sup>x</sup>*2) ∩ ... ∩ N <sup>−</sup>(*xs*) = ∅*. The degree of membership of hyperedge E* = {*<sup>x</sup>*1, *x*2, ... , *xs*} *is defined as*

$$\begin{split} \lambda\_d(E) &= [\mu(\mathbf{x}\_1) \wedge \mu(\mathbf{x}\_2) \wedge \dots \wedge \mu(\mathbf{x}\_s)] \times \\ &\quad [h(\mathcal{N}^+(\mathbf{x}\_1) \cap \mathcal{N}^+(\mathbf{x}\_2) \cap \dots \cap \mathcal{N}^+(\mathbf{x}\_s)) \wedge h(\mathcal{N}^-(\mathbf{x}\_1) \cap \mathcal{N}^-(\mathbf{x}\_2) \cap \dots \cup \mathcal{N}^-(\mathbf{x}\_s))]. \end{split}$$

The method for the construction of fuzzy double competition hypergraph is given in Algorithm A4.

**Lemma 2.** *The fuzzy double competition hypergraph is the intersection of fuzzy row hypergraph and fuzzy column hypergraph.*

**Proof.** Let #» *G* = (*μ*, #» *λ*) be a fuzzy digraph; then, for any hyperedge *E* = {*<sup>x</sup>*1, *x*2,..., *xs*} of CH(#»*G*),

*<sup>λ</sup>d*(*E*) =[*μ*(*<sup>x</sup>*1) ∧ *μ*(*<sup>x</sup>*2) ∧ ... ∧ *μ*(*xs*)]× [*h*(N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xs*)) ∧ *h*(N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xs*))]. =[*μ*(*<sup>x</sup>*1) ∧ *μ*(*<sup>x</sup>*2) ∧ ... ∧ *μ*(*xs*)]× [max*j* { #» *<sup>λ</sup>*(*<sup>x</sup>*1*xj*) ∧ #» *<sup>λ</sup>*(*<sup>x</sup>*2*xj*) ∧ ... ∧ #» *<sup>λ</sup>*(*xnxj*)} ∧ max*k* { #» *<sup>λ</sup>*(*xkx*1) ∧ #» *<sup>λ</sup>*(*xkx*2) ∧ ... ∧ #» *<sup>λ</sup>*(*xkxn*)}]. =[{*μ*(*<sup>x</sup>*1) ∧ *μ*(*<sup>x</sup>*2) ∧ ... ∧ *μ*(*xs*)} × max*j* { #» *<sup>λ</sup>*(*<sup>x</sup>*1*xj*) ∧ #» *<sup>λ</sup>*(*<sup>x</sup>*2*xj*) ∧ ... ∧ #» *<sup>λ</sup>*(*xnxj*)}]× [{*μ*(*<sup>x</sup>*1) ∧ *μ*(*<sup>x</sup>*2) ∧ ... ∧ *μ*(*xs*)} × max*k* { #» *<sup>λ</sup>*(*xkx*1) ∧ #» *<sup>λ</sup>*(*xkx*2) ∧ ... ∧ #» *<sup>λ</sup>*(*xkxn*)}] <sup>=</sup>*λr*(*E*) ∧ *<sup>λ</sup>cl*(*E*).

It follows that the fuzzy double competition hypergraph is the intersection of a fuzzy row hypergraph and fuzzy column hypergraph.

**Example 3.** *Consider the example of fuzzy digraph shown in Figure 1. From Example 2, the fuzzy double competition hypergraph of Figure 1 is given in Figure 6. In addition, Figures 2, 3 and 6 show that the fuzzy double competition hypergraph is the intersection of fuzzy row hypergraph and fuzzy column hypergraph.*

**Figure 6.** DCH(#»*G*).

**Definition 9.** *Let* #» *G* = (*μ*, #» *λ*) *be a fuzzy digraph on a non-empty set X. The fuzzy niche hypergraph* N <sup>H</sup>(#»*G*)=(*μ*, *<sup>λ</sup>n*) *has the same vertex set as* #»*G and there is hyperedge consisting of vertices x*1, *x*2, ... , *xs if either* N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xs*) = ∅ *or* N <sup>−</sup>(*<sup>x</sup>*1) ∩ N <sup>−</sup>(*<sup>x</sup>*2) ∩ ... ∩ N <sup>−</sup>(*xs*) = ∅*. The degree of membership of hyperedge E* = {*<sup>x</sup>*1, *x*2,..., *xs*} *is defined as*

$$\begin{split} \lambda\_{\mathsf{H}}(E) &= \left[ \mu(\mathbf{x}\_{1}) \wedge \mu(\mathbf{x}\_{2}) \wedge \dots \wedge \mu(\mathbf{x}\_{\mathsf{s}}) \right] \times \\ & \left[ h\left( \mathcal{N}^{+}(\mathbf{x}\_{1}) \cap \mathcal{N}^{+}(\mathbf{x}\_{2}) \cap \dots \cap \mathcal{N}^{+}(\mathbf{x}\_{\mathsf{s}}) \right) \vee h\left( \mathcal{N}^{-}(\mathbf{x}\_{1}) \cap \mathcal{N}^{-}(\mathbf{x}\_{2}) \cap \dots \cap \mathcal{N}^{-}(\mathbf{x}\_{\mathsf{s}}) \right) \right]. \end{split}$$

**Lemma 3.** *The fuzzy niche hypergraph is the union of fuzzy row hypergraph and fuzzy column hypergraph.*

**Example 4.** *The fuzzy niche hypergraph of Figure 1 is shown in Figure 7, which is the union of Figures 2 and 3.*

**Figure 7.** N <sup>H</sup>(#»*G*).

**Definition 10.** *Let H be a fuzzy hypergraph and t be the smallest non-negative number such that H* ∪ *It is a fuzzy niche hypergraph of some fuzzy digraph* #» *G*, *where It is a fuzzy set on t isolated vertices Xt; then, t is called fuzzy niche number of H denoted by* <sup>n</sup>(*H*)*.*

**Lemma 4.** *Let H be a fuzzy hypergraph on a non-empty set X with* n(*H*) = *t* < ∞ *and H* ∪ *It is a fuzzy niche hypergraph of an acyclic digraph* #» *G then for all, x* ∈ *X* ∪ *Xt,*

$$\begin{aligned} \mathcal{N}^+(\boldsymbol{y}) \cap I\_t \neq \mathcal{O} &\Rightarrow \exists \; z \in \operatorname{supp}(I\_t) \; \\ \mathcal{N}^-(\boldsymbol{y}) \cap I\_t \neq \mathcal{O} &\Rightarrow \exists \; z \in \operatorname{supp}(I\_t) \; \\ \end{aligned} \text{ such that } \operatorname{supp}(\mathcal{N}^+(\boldsymbol{y})) = \boldsymbol{z}.$$

**Proof.** On the contrary, assume that, for some *y* ∈ *X*, either *supp*(N +(*y*)) = {*z*} ∪ *X* or *supp*(N <sup>−</sup>(*y*)) = {*z*} ∪ *X* , where ∅ = *X* ⊆ *X* ∪ *Xt* \ {*z*}. Then, by definition of a fuzzy niche hypergraph, *z* is adjacent to all vertices *X* in *H* ∪ *It*—a contradiction to the fact that *z* ∈ *Xt*.

**Lemma 5.** *Let H be a fuzzy hypergraph with* n(*H*) = *t* < ∞ *and H* ∪ *It is a fuzzy niche hypergraph of an acyclic fuzzy digraph* #» *G then for all z* ∈ *Xt,* N +(*z*) = ∅ *and* N <sup>−</sup>(*z*) = ∅*.*

**Proof.** On the contrary, assume that *<sup>X</sup>*+*z* = {*y*1, *y*2, ... , *ys*} and *X*<sup>−</sup>*z* = {*y* 1, *y* 2, ... , *y r*}. Clearly, N +(*z*) ∩ N <sup>−</sup>(*z*) = ∅ because #»*G* is acyclic. According to Lemma 4, N +(*yi*) = N +(*y i*). Consider another fuzzy digraph #» *G* such that *<sup>X</sup>*#»*G* = *<sup>X</sup>*#»*G*\ {*z*} and *<sup>E</sup>*#»*G* = (*<sup>E</sup>*#»*G*\ {*<sup>E</sup>*1}) ∪ *E*2, where

$$\begin{aligned} E\_1 &= \{ \overrightarrow{zy\_i} : 1 \le i \le s \} \cup \{ \overrightarrow{y\_i'} \hat{z} : 1 \le i \le r \}, \\ E\_2 &= \{ \overrightarrow{y\_1'y\_i} : 1 \le i \le s \} \cup \{ \overrightarrow{y\_i'y\_1} : 1 \le i \le r \}. \end{aligned}$$

Clearly, N +(*z*) = N +(*<sup>y</sup>*1) and N <sup>−</sup>(*z*) = N <sup>−</sup>(*y* 1). Thus, N H(#»*G* ) = *H* ∪ *It*−<sup>1</sup> which contradicts the fact that n(*H*) = *t*. Hence, for all *z* ∈ *Xt*, N +(*z*) = ∅ and N <sup>−</sup>(*z*) = ∅.

**Definition 11.** *Let H* = (*μ*, ρ) *be a fuzzy hypegraph on a non-empty set X. A hyperedge Ei* = {*<sup>x</sup>*1, *x*2,..., *xr*} ⊆ *X is called strong if* ρ(*Ei*) ≥ 12 D*r k*=1 *μi*(*xk*)*.*

**Theorem 1.** *Let* #» *G* = (*μ*, #» *λ*) *be a fuzzy digraph. If* N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*) *contains exactly one vertex, then the hyperedge* {*<sup>x</sup>*1, *x*2, ... , *xr*} *of* C(#»*G*) *is strong if and only if* |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*)| > 12 *.*

**Proof.** Assume that N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*) = {(*<sup>u</sup>*, *l*)}, where *l* is degree of membership of *u*. As |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*)| = *l* = *h*(N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*)); therefore, *<sup>λ</sup>c*({*<sup>x</sup>*1, *x*2, ... , *xr*})=(*μ*(*<sup>x</sup>*1) ∧ *μ*(*<sup>x</sup>*2) ∧ ... ∧ *μ*(*xr*)) × *h*(N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*)) = *l* × (*μ*(*<sup>x</sup>*1) ∧ *μ*(*<sup>x</sup>*2) ∧ ... ∧ *μ*(*xr*)}). Thus, the hyperedge {*<sup>x</sup>*1, *x*2, ... , *xr*} in C(#»*G*) would be strong if *l* > 12 by Definition 11.

**Definition 12.** *Let k be a non-negative real number number; then, the fuzzy k-competition hypergraph of a fuzzy digraph* #» *G* = (*μ*, #» *λ*) *is fuzzy hypergraph* <sup>C</sup>*k*(#»*G*)=(*μ*, *<sup>λ</sup>kc*), *which has the same fuzzy vertex set as in* #» *G and there is a hyperedge E* = {*<sup>x</sup>*1, *x*2, ... , *xr*} *in* <sup>C</sup>*k*(#»*G*) *if* |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*)| > *k. The membership degree of the hyperedge E is defined as*

$$\lambda\_{\mathbf{k}\mathbf{r}}(\mathbf{E}) = \frac{l-k}{l} (\mu(\mathbf{x}\_1) \wedge \mu(\mathbf{x}\_2) \wedge \dots \wedge \mu(\mathbf{x}\_r)) \times h(\mathcal{N}^+(\mathbf{x}\_1) \cap \mathcal{N}^+(\mathbf{x}\_2) \cap \dots \cap \mathcal{N}^+(\mathbf{x}\_r)),$$

*where* |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*)| = *l.*

**Example 5.** *The fuzzy* 0.2−*competition hypergraph of Figure 1 is given in Figure 8.*

**Figure 8.** Fuzzy 0.2−competition hypergraph.

**Remark 1.** *For k* = 0*, a fuzzy k-competition hypergraph is simply a fuzzy competition hypergraph.* **Theorem 2.** *Let* #» *G* = (*μ*, #» *λ*) *be a fuzzy digraph. If h*(N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*)) = 1 *and* |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*)| > 2*k for some x*1, *x*2, ... , *xr* ∈ *X, then the hyperedge* {*<sup>x</sup>*1, *x*2, ... , *xr*} *is strong in* <sup>C</sup>*k*(#»*G*)*.*

**Proof.** Let <sup>C</sup>*k*(#»*G*)=(*μ*, *<sup>λ</sup>kc*) be a fuzzy *k*-competition hypergraph of fuzzy digraph #»*G* = (*μ*, #»*<sup>λ</sup>*). Suppose for *E* = {*<sup>x</sup>*1, *x*2,..., *xr*} ⊆ *X*, |N +(*<sup>x</sup>*1) ∩ N +(*<sup>x</sup>*2) ∩ ... ∩ N +(*xr*)| = *l*. Now,

$$\begin{split} \lambda\_{\text{kr}}(E) &= \frac{l-k}{l} (\mu(\mathbf{x}\_1) \wedge \mu(\mathbf{x}\_2) \wedge \dots \wedge \mu(\mathbf{x}\_r)) \times h(\mathcal{N}^+(\mathbf{x}\_1) \cap \mathcal{N}^+(\mathbf{x}\_2) \cap \dots \cap \mathcal{N}^+(\mathbf{x}\_r)), \\ \lambda\_{\text{kr}}(E) &= \frac{l-k}{l} (\mu(\mathbf{x}\_1) \wedge \mu(\mathbf{x}\_2) \wedge \dots \wedge \mu(\mathbf{x}\_r)), \quad \vdots \, h(\mathcal{N}^+(\mathbf{x}\_1) \cap \mathcal{N}^+(\mathbf{x}\_2) \cap \dots \cap \mathcal{N}^+(\mathbf{x}\_r)) = 1, \\ &\xrightarrow{\lambda\_{\text{kr}}(E)} \frac{\lambda\_{\text{kr}}(E)}{\mu(\mathbf{x}\_1) \wedge \mu(\mathbf{x}\_2) \wedge \dots \wedge \mu(\mathbf{x}\_r)} > \frac{1}{2}, \qquad \because \; l > 2k. \end{split}$$

Thus, the hyperedge *E* is strong in <sup>C</sup>*k*(#»*G*).
