Directed hyperpath

Directed hyperpath --- ориентированный гиперпуть.

A directed hyperpath [math]\displaystyle{ P_{Rt} }[/math] from the root set [math]\displaystyle{ R \; (\subseteq V) }[/math] to the sink [math]\displaystyle{ t \; (\in V) }[/math] in [math]\displaystyle{ H }[/math] is a minimal acyclic sub-hypergraph of [math]\displaystyle{ H }[/math] containing both the nodes of [math]\displaystyle{ R }[/math] and node [math]\displaystyle{ t }[/math], such that each node, with exception of the nodes in [math]\displaystyle{ R }[/math], has exactly one entering hyperarc.

The definition of a hyperpath can be extended as follows. A (directed) hypertree [math]\displaystyle{ T_{R} }[/math] rooted at [math]\displaystyle{ R }[/math] in [math]\displaystyle{ H }[/math] is an acyclic sub-hypergraph of [math]\displaystyle{ H }[/math] containing the nodes in [math]\displaystyle{ R }[/math], such that each node, with the exception of the nodes in [math]\displaystyle{ R }[/math] has exactly one entering hyperarc.

The set [math]\displaystyle{ R }[/math] is called the root set, while the remaining nodes are called the non-roots. Any non-root [math]\displaystyle{ v }[/math] not contained in the tail of any hyperarc of [math]\displaystyle{ T_{R} }[/math] is said to be a leaf of the hypertree. By definition, for any non-root [math]\displaystyle{ v }[/math] there is a unique directed hyperpath in [math]\displaystyle{ T_{R} }[/math] from [math]\displaystyle{ R }[/math] to [math]\displaystyle{ v }[/math].

An undirected hyperpath (hypertree) is a permutation of a hyperpath [math]\displaystyle{ P_{Rt} }[/math] (hypertree [math]\displaystyle{ T_{R} }[/math]), i.e. it is obtained by a permutation of some of the hyperarcs on [math]\displaystyle{ P_{Rt} }[/math] ([math]\displaystyle{ T_{R} }[/math]), where the permutation of a hyperarc [math]\displaystyle{ e }[/math] is a hyperarc [math]\displaystyle{ e' }[/math] such that [math]\displaystyle{ T_{e}' \cup h_{e}' = T_{e} \sup h_{e} }[/math].