Directed hypergraph

Directed hypergraph --- ориентированный гиперграф.

A directed hypergraph [math]\displaystyle{ H }[/math] is a pair [math]\displaystyle{ (N,E) }[/math], where [math]\displaystyle{ N }[/math] is a nonempty set of nodes or vertices and [math]\displaystyle{ E }[/math] is a set of hyperarcs; a hyperarc [math]\displaystyle{ e }[/math] is an ordered pair [math]\displaystyle{ (T,h) }[/math], with [math]\displaystyle{ T \subseteq N, \; T \neq \emptyset }[/math] and [math]\displaystyle{ h \in N }[/math]; [math]\displaystyle{ h }[/math] and [math]\displaystyle{ T }[/math] are called the head and the tail of the hyperarc [math]\displaystyle{ e }[/math] and will be denoted with [math]\displaystyle{ Head(e) }[/math] and [math]\displaystyle{ Tail(e) }[/math], respectively.

The size of a directed hypergraph can be defined as a sum of the cardinalities of its hyperarcs:

[math]\displaystyle{ size(H) = \sum_{e \in E}|T_{e}|. }[/math]