Niche hypergraphs

If D = (V,A) is a digraph, its niche hypergraph Nℋ(D) = (V, ℰ) has the edge set ℰ = {ℯ ⊆ V | |e| ≥ 2 ∧ ∃ v ∈ V : e = ND-(v) ∨ ℯ = ND+(v)}. Niche hypergraphs generalize the well-known niche graphs (see [11]) and are closely related to competition hypergraphs (see [40]) as well as double competition hypergraphs (see [33]). We present several properties of niche hypergraphs of acyclic digraphs

An algorithm to compute the strength of competing interactions in the Bering Sea based on pythagorean fuzzy hypergraphs

[EN] The networks of various problems have competing constituents, and there is a concern to compute the strength of competition among these entities. Competition hypergraphs capture all groups of predators that are competing in a community through their hyperedges. This paper reintroduces competition hypergraphs in the context of Pythagorean fuzzy set theory, thereby producing Pythagorean fuzzy competition hypergraphs. The data of real-world ecological systems posses uncertainty, and the proposed hypergraphs can efficiently deal with such information to model wide range of competing interactions. We suggest several extensions of Pythagorean fuzzy competition hypergraphs, including Pythagorean fuzzy economic competition hypergraphs, Pythagorean fuzzy row as well as column hypergraphs, Pythagorean fuzzy k-competition hypergraphs, m-step Pythagorean fuzzy competition hypergraphs and Pythagorean fuzzy neighborhood hypergraphs. The proposed graphical structures are good tools to measure the strength of direct and indirect competing and non-competing interactions. Their aptness is illustrated through examples, and results support their intrinsic interest. We propose algorithms that help to compose some of the presented graphical structures. We consider predator-prey interactions among organisms of the Bering Sea as an application: Pythagorean fuzzy competition hypergraphs encapsulate the competing relationships among its inhabitants. Specifically, the algorithm which constructs the Pythagorean fuzzy competition hypergraphs can also compute the strength of competing and non-competing relations of this scenario.Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature.Publicación en abierto financiada por el Consorcio de Bibliotecas Universitarias de Castilla y León (BUCLE), con cargo al Programa Operativo 2014ES16RFOP009 FEDER 2014-2020 DE CASTILLA Y LEÓN, Actuación:20007-CL - Apoyo Consorcio BUCL

Niche Hypergraphs

If D = (V,A) is a digraph, its niche hypergraph NH(D) = (V, E) has the edge set ℇ = {e ⊆ V | |e| ≥ 2 ∧ ∃ v ∈ V : e = N−D(v) ∨ e = N+D(v)}. Niche hypergraphs generalize the well-known niche graphs (see [11]) and are closely related to competition hypergraphs (see [40]) as well as double competition hypergraphs (see [33]). We present several properties of niche hypergraphs of acyclic digraphs

Niche Hypergraphs of Products of Digraphs

If D = (V, A) is a digraph, its niche hypergraph Nℋ(D) = (V, ℰ) has the edge set ℰ={e⊆V||e|≥2∧∃ υ∈V:e=ND−(υ)∨e=ND+(υ)}{\cal E} = \{ {e \subseteq V| | e | \ge 2 \wedge \exists \, \upsilon \in V:e = N_D^ - ( \upsilon ) \vee e = N_D^ + ( \upsilon )} \} . Niche hypergraphs generalize the well-known niche graphs and are closely related to competition hypergraphs as well as common enemy hypergraphs. For several products D1 ◦ D2 of digraphs D1 and D2, we investigate the relations between the niche hypergraphs of the factors D1, D2 and the niche hypergraph of their product D1 ◦ D2

Niche Hypergraphs of Products of Digraphs

If $D = (V, A)$ is a digraph, its niche hypergraph $N \mathcal{H} (D) = (V, \mathcal{E} )$ has the edge set $\mathcal{E} = \{ e \subseteq V \ | \ |e| \le 2 \land \exists υ \in V : e = N_D^− (υ) \lor e=N_D^+ (υ) \}$. Niche hypergraphs generalize the well-known niche graphs and are closely related to competition hypergraphs as well as common enemy hypergraphs. For several products $D_1 \circ D_2$ of digraphs $D_1$ and $D_2$, we investigate the relations between the niche hypergraphs of the factors $D_1$, $D_2$ and the niche hypergraph of their product $D_1 \circ D_2$

A note on a conjecture on niche hypergraphs

summary:For a digraph $D$, the niche hypergraph $N\mathcal {H}(D)$ of $D$ is the hypergraph having the same set of vertices as $D$ and the set of hyperedges $E(N\mathcal {H}(D)) = \{e \subseteq V(D) \colon |e| \geq 2$ and there exists a vertex $v$ such that $e = N^{-}_{D}(v)$ or $e = N^{+}_{D}(v)\}$. A digraph is said to be acyclic if it has no directed cycle as a subdigraph. For a given hypergraph $\mathcal {H}$, the niche number $\hat {n}(\mathcal {H})$ is the smallest integer such that $\mathcal {H}$ together with $\hat {n}(\mathcal {H})$ isolated vertices is the niche hypergraph of an acyclic digraph. C. Garske, M. Sonntag and H. M. Teichert (2016) conjectured that for a linear hypercycle $\mathcal {C}_{m}$, $m \geq 2$, if $\min \{|e| \colon e \in E(\mathcal {C}_{m})\} \geq 3$, then $\hat {n}(\mathcal {C}_{m}) = 0$. In this paper, we prove that this conjecture is true