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In [[mathematics]], a '''hypergraph''' is a generalization of a [[Graph (discrete mathematics)|graph]] in which an [[graph theory|edge]] can join any number of [[vertex (graph theory)|vertices]]. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a hypergraph <math>H</math> is a pair <math>H = (X,E)</math> where <math>X</math> is a set of elements called ''nodes'' or ''vertices'', and <math>E</math> is a set of non-empty subsets of <math>X</math> called ''[[hyperedges]]'' or ''edges''. Therefore, <math>E</math> is a subset of <math>\mathcal{P}(X) \setminus\{\emptyset\}</math>, where <math>\mathcal{P}(X)</math> is the [[power set]] of <math>X</math>. The size of the vertex set is called the ''order of the hypergraph'', and the size of edges set is the ''size of the hypergraph''.  

{{Short description|Generalization of graph theory}}In [[mathematics]], a '''hypergraph''' is a generalization of a [[Graph (discrete mathematics)|graph]] in which an [[graph theory|edge]] can join any number of [[vertex (graph theory)|vertices]]. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a hypergraph <math>H</math> is a pair <math>H = (X,E)</math> where <math>X</math> is a set of elements called ''nodes'' or ''vertices'', and <math>E</math> is a set of non-empty subsets of <math>X</math> called ''[[hyperedges]]'' or ''edges''. Therefore, <math>E</math> is a subset of <math>\mathcal{P}(X) \setminus\{\emptyset\}</math>, where <math>\mathcal{P}(X)</math> is the [[power set]] of <math>X</math>. The size of the vertex set is called the ''order of the hypergraph'', and the size of edges set is the ''size of the hypergraph''.  

While graph edges are 2-element subsets 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, one such hypergraph is a collection of sets, each such set a hyperedge connecting ''k'' nodes.) 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]].  

While graph edges are 2-element subsets 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, one such hypergraph is a collection of sets, each such set a hyperedge connecting ''k'' nodes.) 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]].