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==Symmetric hypergraphs==

==Symmetric hypergraphs==

The<math>r(H)</math> of a hypergraph <math>H</math> is the maximum cardinality of any of the edges in the hypergraph.  If all edges have the same cardinality ''k'', the hypergraph is said to be ''uniform'' or ''k-uniform'', or is called a ''k-hypergraph''.  A graph is just a 2-uniform hypergraph.

The<math>r(H)</math> of a hypergraph <math>H</math> is the maximum cardinality of any of the edges in the hypergraph.  If all edges have the same cardinality ''k'', the hypergraph is said to be ''uniform'' or ''k-uniform'', or is called a ''k-hypergraph''.  A graph is just a 2-uniform hypergraph.

超图<math>H</math>的<math>r(H)</math>表示该超图中任何一条边的最大基数。如果所有边具有相同的基数''k''，则称该超图为''均匀的''或''k-均匀的''，或称之为''k-超图''。图只是一个2-均匀的超图。

超图<math>H</math>的<math>r(H)</math>表示该超图中任何一条边的最大基数。如果所有边具有相同的基数''k''，则称该超图为''均匀的''或''k-均匀的''，或称之为''k-超图''。图只是一个2-均匀的超图。

The degree ''d(v)'' of a vertex ''v'' is the number of edges that contain it. ''H'' is ''k-regular'' if every vertex has degree ''k''.

The degree ''d(v)'' of a vertex ''v'' is the number of edges that contain it. ''H'' is ''k-regular'' if every vertex has degree ''k''.

顶点''v''的度''d(v)''表示包含该顶点的边的数量。如果每个顶点的度都为''k''，则超图''H''是''k-正则的''。

顶点''v''的度''d(v)''表示包含该顶点的边的数量。如果每个顶点的度都为''k''，则超图''H''是''k-正则的''。

The dual of a uniform hypergraph is regular and vice versa.

The dual of a uniform hypergraph is regular and vice versa.

均匀超图的对偶是正则的，反之亦然。

均匀超图的对偶是正则的，反之亦然。

Two vertices ''x'' and ''y'' of ''H'' are called ''symmetric'' if there exists an automorphism such that <math>\phi(x)=y</math>.  Two edges <math>e_i</math> and <math>e_j</math> are said to be  ''symmetric'' if there exists an automorphism such that <math>\phi(e_i)=e_j</math>.

Two vertices ''x'' and ''y'' of ''H'' are called ''symmetric'' if there exists an automorphism such that <math>\phi(x)=y</math>.  Two edges <math>e_i</math> and <math>e_j</math> are said to be  ''symmetric'' if there exists an automorphism such that <math>\phi(e_i)=e_j</math>.

如果存在一个形如<math>\phi(x)=y</math>的自同构，则超图''H''的两个顶点''x''和''y''''对称''。如果存在一个自同构使得<math>\phi(e_i)=e_j</math>，则称两个边<math>e_i</math>和<math>e_j</math>为''对称''。

如果存在一个形如<math>\phi(x)=y</math>的自同构，则超图''H''的两个顶点''x''和''y''''对称''。如果存在一个自同构使得<math>\phi(e_i)=e_j</math>，则称两个边<math>e_i</math>和<math>e_j</math>为''对称''。

A hypergraph is said to be ''vertex-transitive'' (or ''vertex-symmetric'') if all of its vertices are symmetric. Similarly, a hypergraph is ''edge-transitive'' if all edges are symmetric. If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply ''transitive''.

A hypergraph is said to be ''vertex-transitive'' (or ''vertex-symmetric'') if all of its vertices are symmetric. Similarly, a hypergraph is ''edge-transitive'' if all edges are symmetric. If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply ''transitive''.

如果超图的所有顶点都是对称的，则称其为''顶点可传递的''(或''顶点对称的'')。类似地，如果超图的所有边都是对称的，则该超图是''边传递的''。 如果一个超图既是边对称的又是顶点对称的，则该超图是''简单传递的''。

如果超图的所有顶点都是对称的，则称其为''顶点可传递的''(或''顶点对称的'')。类似地，如果超图的所有边都是对称的，则该超图是''边传递的''。 如果一个超图既是边对称的又是顶点对称的，则该超图是''简单传递的''。

Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity.

Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity.

由于超图的对偶性，边传递性的研究与顶点传递性的研究是相一致的。

由于超图的对偶性，边传递性的研究与顶点传递性的研究是相一致的。

==Transversals==

==Transversals==

A ''[[Transversal (combinatorics)|transversal]]'' (or "[[hitting set]]") of a hypergraph ''H'' = (''X'', ''E'') is a set <math>T\subseteq X</math> that has nonempty [[intersection (set theory)|intersection]] with every edge. A transversal ''T'' is called ''minimal'' if no proper subset of ''T'' is a transversal. The ''transversal hypergraph'' of ''H'' is the hypergraph (''X'', ''F'') whose edge set ''F'' consists of all minimal transversals of ''H''.

A ''[[Transversal (combinatorics)|transversal]]'' (or "[[hitting set]]") of a hypergraph ''H'' = (''X'', ''E'') is a set <math>T\subseteq X</math> that has nonempty [[intersection (set theory)|intersection]] with every edge. A transversal ''T'' is called ''minimal'' if no proper subset of ''T'' is a transversal. The ''transversal hypergraph'' of ''H'' is the hypergraph (''X'', ''F'') whose edge set ''F'' consists of all minimal transversals of ''H''.

超图''H'' = (''X'', ''E'')的横截集（或“命中集”）是一个<math>T\subseteq X</math>集合，该集合与每条边都有非空的交集。如果''T''的真子集不是横截集，则称''T''为极小截集。''H'' 的横截超图是超图（''X''， ''F''），其边集''F''包含''H''的所有最小横截。

超图''H'' = (''X'', ''E'')的横截集（或“命中集”）是一个<math>T\subseteq X</math>集合，该集合与每条边都有非空的交集。如果''T''的真子集不是横截集，则称''T''为极小截集。''H'' 的横截超图是超图（''X''， ''F''），其边集''F''包含''H''的所有最小横截。

Computing the transversal hypergraph has applications in [[combinatorial optimization]], in [[game theory]], and in several fields of [[computer science]] such as [[machine learning]], [[Index (database)|indexing of database]]s, the [[Boolean satisfiability problem|satisfiability problem]], [[data mining]], and computer [[program optimization]].

Computing the transversal hypergraph has applications in [[combinatorial optimization]], in [[game theory]], and in several fields of [[computer science]] such as [[machine learning]], [[Index (database)|indexing of database]]s, the [[Boolean satisfiability problem|satisfiability problem]], [[data mining]], and computer [[program optimization]].

计算横向超图在[[组合优化 Combinatorial Optimization]]、[[博弈论 Game Theory]]和[[计算机科学 Computer Science]]的一些领域（例如[[机器学习 Machine Learning]]、[[数据库索引 Indexing of Databases]]、[[可满足性问题the Satisfiability Problem]]、[[数据挖掘Data Mining]]和[[计算机程序优化 Program Optimization]]）都有应用。

 

计算横截面超图在[[组合优化 Combinatorial Optimization]]、[[博弈论 Game Theory]]和[[计算机科学 Computer Science]]的一些领域（例如[[机器学习 Machine Learning]]、[[数据库索引 Indexing of Databases]]、[[可满足性问题the Satisfiability Problem]]、[[数据挖掘Data Mining]]和[[计算机程序优化 Program Optimization]]）都有应用。

 

==Incidence matrix==

==Incidence matrix==

Let <math>V = \{v_1, v_2, ~\ldots, ~ v_n\}</math> and <math>E = \{e_1, e_2, ~ \ldots ~ e_m\}</math>. Every hypergraph has an <math>n \times m</math> [[incidence matrix]] <math>A = (a_{ij})</math> where

Let <math>V = \{v_1, v_2, ~\ldots, ~ v_n\}</math> and <math>E = \{e_1, e_2, ~ \ldots ~ e_m\}</math>. Every hypergraph has an <math>n \times m</math> [[incidence matrix]] <math>A = (a_{ij})</math> where