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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''.  

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''.  

在[[数学中]], '''超图 hypergraph'''是有限集合的子集系统，是最一般的离散结构，在信息科学、生命科学等领域有着广泛的应用。它的一条[[graph theory|边]]可以连接任意数量的[[vertex (graph theory)|顶点]]. 相对而言，在普通图中，一条边只能连接两个顶点。形式上, 超图 <math>H</math> 是一个集合组 <math>H = (X,E)</math> 其中<math>X</math> 是一个以节点或顶点为元素的集合，即顶点集, 而 <math>E</math> 是一组非空子集，被称为边或超边.

在[[数学中]], '''超图 hypergraph'''是有限集合的子集系统，是最一般的离散结构，在信息科学、生命科学等领域有着广泛的应用。它的一条'''边 edge'''可以连接任意数量的'''顶点 vertices'''。相对而言，在普通图中，一条边只能连接两个顶点。形式上，超图 <math>H</math> 是一个有序二元组 <math>H = (X,E)</math> 其中<math>X</math> 是一个以节点 nodes或顶点为元素的非空集合，即顶点集，而 <math>E</math> 是<math>X</math> 的一组非空子集簇，<math>E</math>的元素被称为边或超边 hyperedges。 没有相同边的超图称为单超图。

因此，若<math>\mathcal{P}(X)</math>是 <math>E</math>的幂集，则<math>X</math>是 <math>\mathcal{P}(X) \setminus\{\emptyset\}</math> 的一个子集。在<math>H</math>中，顶点集的大小被称为超图的阶数，边集的大小被称为超图的大小。

 

因此，若<math>\mathcal{P}(X)</math>是 <math>E</math>的幂集 power set，则<math>X</math>是 <math>\mathcal{P}(X) \setminus\{\emptyset\}</math> 的一个子集。在<math>H</math>中，顶点集的大小被称为'''超图的阶数 order of the hypergraph'''，边集的大小被称为'''超图的大小 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]].  

普通图的边是节点的二元子集，超边则是节点的任意集合，所以可以包含任意数量的节点。但我们总先需要研究具有相同基数超边的超图，即一个 k-均匀超图，所有超边的大小都为 k。因此一个 2-均匀超图就是图，一个 3-均匀超图就是三元组的集合，依此类推。超图也被称为从[[泛集]]（universal set）中抽取的一个集系统或[[集族]]。

普通图的边是节点的二元子集，超边则是节点的任意集合，所以可以包含任意数量的节点。但我们总先需要研究具有相同基数超边的超图，即一个'''k-均匀超图 k-uniform hypergraph'''（所有超边的大小都为 k）。因此一个2-均匀超图就是普通图，一个3-均匀超图就是无序三元组 unordered triples 的集合，依此类推。超图也被称为从泛集 universal set 中抽取的一个集系统或集族 universal set。

Hypergraphs can be viewed as [[incidence structure]]s.  In particular, there is a bipartite "incidence graph" or "[[Levi graph]]" corresponding to every hypergraph, and conversely, most, but not all, [[bipartite graph]]s can be regarded as incidence graphs of hypergraphs.

Hypergraphs can be viewed as [[incidence structure]]s.  In particular, there is a bipartite "incidence graph" or "[[Levi graph]]" corresponding to every hypergraph, and conversely, most, but not all, [[bipartite graph]]s can be regarded as incidence graphs of hypergraphs.

超图有不同的类型，如：

超图有不同的类型，如：

* 空超图：没有边的超图

* 空超图 Empty hypergraph：没有边的超图

* 非简单（或多重）超图：允许有循环（有单个顶点的超边）或重复边的超图，也就是说可以有两个或两个以上的边包含同一组顶点。

* 非简单（或多重）超图：允许有循环（有单个顶点的超边）或重复边的超图，也就是说可以有两个或两个以上的边包含同一组顶点。

* 简单超图：不包含循环和重复边的超图。

* 简单超图：不包含循环和重复边的超图。