Bihypergraph

Bihypergraph --- бигиперграф.

Let [math]\displaystyle{ H^{0} }[/math] and [math]\displaystyle{ H^{1} }[/math] be hypergraphs with the same vertex set [math]\displaystyle{ V }[/math]. An ordered pair [math]\displaystyle{ H = (H^{0},H^{1}) }[/math] is called a bihypergraph with the set of 0-edges [math]\displaystyle{ E(H^{0}) }[/math] and the set of 1-edges [math]\displaystyle{ E(H^{1}) }[/math]. Every hyperedge of either [math]\displaystyle{ H^{0} }[/math] or [math]\displaystyle{ H^{1} }[/math] is considered as a hyperedge of [math]\displaystyle{ H }[/math]. The order of [math]\displaystyle{ H }[/math] is [math]\displaystyle{ n(H) = |V| }[/math]. The rank of [math]\displaystyle{ H }[/math] is [math]\displaystyle{ r(H) = \max\{r(H^{0}), r(H^{1})\} }[/math].

A bihypergraph [math]\displaystyle{ H = (H^{0}, H^{1}) }[/math] is called bipartite if there exists an ordered partition [math]\displaystyle{ V^{0} \cup V^{1} = V(H) }[/math] (bipartition) such that the set [math]\displaystyle{ V^{i} }[/math] is stable in [math]\displaystyle{ H^{i} }[/math], [math]\displaystyle{ i = 0,1 }[/math].