Connected component of a hypergraph

Connected component of a hypergraph --- связная компонента гиперграфа.

Let [math]\displaystyle{ {\mathcal E} = (V, \{E_{1}, \ldots, E_{m}\}) }[/math] be a hypergraph. A sequence [math]\displaystyle{ (E_{1}, \ldots, E_{k}) }[/math] of distinct hyperedges is a path of length [math]\displaystyle{ k }[/math] if for all [math]\displaystyle{ i, \; 1 \leq i \lt m, \; E_{i} \cap E_{i+1} \neq \emptyset }[/math]. Two vertices [math]\displaystyle{ x \in E_{1}, \; y \in E_{k} }[/math] are connected (by the path [math]\displaystyle{ (E_{1}, \ldots, E_{k}) }[/math]), and [math]\displaystyle{ E_{1} }[/math] and [math]\displaystyle{ E_{k} }[/math] are also connected. A set of hyperedges is connected if every pair of hyperedges in the set is connected. A connected component of a hypergraph is a maximal connected set of hyperedges.