In mathematics, a hypergraph is a generalization of a graph in which an edge can connect any number of vertices. Formally, a hypergraph is a pair where is a set of elements called nodes or vertices, and is a set of non-empty subsets of called hyperedges or edges. Therefore, is a subset of, where is the power set of .
While graph edges are pairs of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k. (In other words, it is a collection of sets of size k.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on.
A hypergraph is also called a set system or a family of sets drawn from the universal set X. The difference between a set system and a hypergraph (which is not well defined) is in the questions being asked. Hypergraph theory tends to concern questions similar to those of graph theory, such as connectivity and colorability, while the theory of set systems tends to ask non-graph-theoretical questions, such as those of Sperner theory.
There are variant definitions; sometimes edges must not be empty, and sometimes multiple edges, with the same set of nodes, are allowed.
Hypergraphs can be viewed as incidence structures. In particular, there is a bipartite "incidence graph" or "Levi graph" corresponding to every hypergraph, and conversely, most, but not all, bipartite graphs can be regarded as incidence graphs of hypergraphs.
Hypergraphs have many other names. In computational geometry, a hypergraph may sometimes be called a range space and then the hyperedges are called ranges. In cooperative game theory, hypergraphs are called simple games (voting games); this notion is applied to solve problems in social choice theory.
Special kinds of hypergraphs include, besides k-uniform ones, clutters, where no edge appears as a subset of another edge; and abstract simplicial complexes, which contain all subsets of every edge.
The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.
Read more about Hypergraph: Terminology, Bipartite Graph Model, Isomorphism and Equality, Symmetric Hypergraphs, Transversals, Incidence Matrix, Hypergraph Coloring, Partitions, Theorems, Hypergraph Drawing, Generalizations