Editing Hypergraph (section)

==Properties of hypergraphs==
A hypergraph can have various properties, such as:
* '''Empty''' - has no edges.
* '''Non-simple''' ''(or'' '''multiple''''')'' - has loops (hyperedges with a single vertex) or repeated edges, which means there can be two or more edges containing the same set of vertices.
* '''Simple''' - has no loops and no repeated edges.
* '''<math>d </math>-regular''' - every vertex has degree <math>d </math>, i.e., contained in exactly <math>d </math> hyperedges.
*'''2-colorable''' - its vertices can be partitioned into two classes ''U'' and ''V'' in such a way that each hyperedge with cardinality at least 2 contains at least one vertex from both classes. An alternative term is '''[[Property B]]'''.
** Two stronger properties are [[Bipartite hypergraph|'''bipartite''']] and [[Balanced hypergraph|'''balanced''']].
* '''<math>k </math>-uniform''' - each hyperedge contains precisely <math>k</math> vertices.
* '''<math>k </math>-partite''' - the vertices are partitioned into <math>k</math> parts, and each hyperedge contains precisely one vertex of each type.
** Every '''<math>k </math>-partite''' hypergraph (for  <math>k\geq 2</math>) is both  '''<math>k </math>'''-uniform and bipartite (and 2-colorable).
* '''Reduced''':<ref>{{Cite journal |last=Fagin |first=Ronald |date=1983-07-01 |title=Degrees of acyclicity for hypergraphs and relational database schemes |journal=Journal of the ACM |volume=30 |issue=3 |pages=514–550 |doi=10.1145/2402.322390 |issn=0004-5411|doi-access=free }}</ref> no hyperedge is a strict subset of another hyperedge; equivalently, every hyperedge is maximal for inclusion. The '''reduction''' of a hypergraph is the reduced hypergraph obtained by removing every hyperedge which is included in another hyperedge.
* '''Downward-closed''' - every subset of an undirected hypergraph's edges is a hyperedge too. A downward-closed hypergraph is usually called an '''[[abstract simplicial complex]]'''. It is generally not reduced, unless all hyperedges have cardinality 1.
** An abstract simplicial complex with the ''augmentation property'' is called a '''[[matroid]]'''.
* '''Laminar''': for any two hyperedges, either they are disjoint, or one is included in the other. In other words, the set of hyperedges forms a [[laminar set family]].