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 \( D_1 \circ D_2 \).