更改

hyperedge of <math>H</math> which is partially contained in the subhypergraph <math>H_A</math> and is fully contained in the extension <math>Ex(H_A)</math>.

hyperedge of <math>H</math> which is partially contained in the subhypergraph <math>H_A</math> and is fully contained in the extension <math>Ex(H_A)</math>.

Formally

Formally

一个''子超图''的''扩展''是一个超图，其中每个属于 <math>H</math> 的超边都部分包含在子超图的 <math>H_A</math>，并且完全包含在扩展的 <math>Ex(H_A)</math> 中。即在形式上：

一个''子超图''的''扩展''是一个超图，其中每个属于 <math>H</math> 的超边都部分包含在子超图的 <math>H_A</math>，并且完全包含在扩展的 <math>Ex(H_A)</math> 中。即在形式上：

:<math>Ex(H_A) = (A \cup A', E' )</math> with <math>A' = \bigcup_{e \in E} e \setminus A</math> and <math>E' = \lbrace e \in E | e \subseteq (A \cup A') \rbrace</math>.

:<math>Ex(H_A) = (A \cup A', E' )</math>和 <math>A' = \bigcup_{e \in E} e \setminus A</math> 和<math>E' = \lbrace e \in E | e \subseteq (A \cup A') \rbrace</math>.

The ''partial hypergraph'' is a hypergraph with some edges removed. Given a subset <math>J \subset I_e</math> of the edge index set, the partial hypergraph generated by <math>J</math> is the hypergraph

The ''partial hypergraph'' is a hypergraph with some edges removed. Given a subset <math>J \subset I_e</math> of the edge index set, the partial hypergraph generated by <math>J</math> is the hypergraph

''部分超图''是去掉一些边的超图。给定一个边索引集的子集 <math>J \subset I_e</math> ，由 <math>J</math> 生成的部分超图就是

''部分超图''是去掉一些边的超图。给定一个边索引集的子集 <math>J \subset I_e</math> ，由 <math>J</math> 生成的部分超图就是

Given a subset <math>A\subseteq X</math>, the ''section hypergraph'' is the partial hypergraph

Given a subset <math>A\subseteq X</math>, the ''section hypergraph'' is the partial hypergraph

而给定一个子集  <math>A\subseteq X</math>，则''分段超图''是部分超图

而给定一个子集  <math>A\subseteq X</math>，则''分段超图''是部分超图

:<math>H \times A = \left(A, \lbrace e_i |  

:<math>H \times A = \left(A, \lbrace e_i |  

The '''dual''' <math>H^*</math> of <math>H</math> is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by <math>\lbrace e_i \rbrace</math> and whose edges are given by <math>\lbrace X_m \rbrace</math> where

The '''dual''' <math>H^*</math> of <math>H</math> is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by <math>\lbrace e_i \rbrace</math> and whose edges are given by <math>\lbrace X_m \rbrace</math> where

<math>H</math> 的重记号 <math>H^*</math> 则是一个顶点和边互换的超图，因此顶点由 <math>\lbrace e_i \rbrace</math> 给出，边由 <math>\lbrace X_m \rbrace</math> 给出，其中

 

 

<math>H</math> 的对偶 <math>H^*</math> 则是一个顶点和边互换的超图，因此顶点由 <math>\lbrace e_i \rbrace</math> 给出，边由 <math>\lbrace X_m \rbrace</math> 给出，其中

:<math>X_m = \lbrace e_i | x_m \in e_i \rbrace. </math>

:<math>X_m = \lbrace e_i | x_m \in e_i \rbrace. </math>

When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an [[involution (mathematics)|involution]], i.e.,

When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an [[involution (mathematics)|involution]], i.e.,

当等式的记号被正确定义时，如下，对一个超图采取两次运算是对偶的：

当等式的记号被正确定义时，如下，对一个超图采取两次运算是对偶的：

:<math>\left(H^*\right)^* = H.</math>

:<math>\left(H^*\right)^* = H.</math>

A [[connected graph]] ''G'' with the same vertex set as a connected hypergraph ''H'' is a '''host graph''' for ''H'' if every hyperedge of ''H'' [[induced subgraph|induces]] a connected subgraph in  ''G''. For a disconnected hypergraph ''H'', ''G'' is a host graph if there is a bijection between the [[connected component (graph theory)|connected components]] of ''G'' and of ''H'', such that each connected component ''G<nowiki>'</nowiki>'' of ''G'' is a host of the corresponding ''H<nowiki>'</nowiki>''.

A [[connected graph]] ''G'' with the same vertex set as a connected hypergraph ''H'' is a '''host graph''' for ''H'' if every hyperedge of ''H'' [[induced subgraph|induces]] a connected subgraph in  ''G''. For a disconnected hypergraph ''H'', ''G'' is a host graph if there is a bijection between the [[connected component (graph theory)|connected components]] of ''G'' and of ''H'', such that each connected component ''G<nowiki>'</nowiki>'' of ''G'' is a host of the corresponding ''H<nowiki>'</nowiki>''.

对于不连通的超图 G 和具有相同顶点连通的超图 H，如果 H 的每个超边都有 G 中一个子图连接，则 G 是一个主图（host graph）；

对于不连通的超图'' G'' 和具有相同顶点连通的超图 ''H''，如果 ''H'' 的每个超边都有 ''G'' 中一个子图连接，则 ''G'' 是一个主图（host graph）；

对于不连通的超图 H，如果 G 和 H 的连通部分之间存在一个双射，使得 G 的每个连通部分 G' 都是对应的 H' 的主图，则 G 是一个主图。

对于不连通的超图 H，如果 ''G'' 和 ''H'' 的连通部分之间存在一个双射，使得 G 的每个连通部分 ''G'' 都是对应的 ''H'' 的主图，则 ''G'' 是一个主图。

A hypergraph is ''bipartite'' if and only if 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. Alternatively, such a hypergraph is said to have [[Property B]].

A hypergraph is ''bipartite'' if and only if 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. Alternatively, such a hypergraph is said to have [[Property B]].

一个超图是二分（bipartite）的，当且仅当它的顶点能被分成两类 U 和 V  ：每个基数至少为 2 超边包含两类中的至少一个顶点。相反的，超图则被称为具有属性 B。

一个超图是二分（bipartite）的，当且仅当它的顶点能被分成两类''U'' 和 ''V'' ：每个基数至少为 2 超边包含两类中的至少一个顶点。相反的，超图则被称为具有[[属性 B]]。

The '''2-section''' (or '''clique graph''', '''representing graph''', '''primal graph''', '''Gaifman graph''') of a hypergraph is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge.

The '''2-section''' (or '''clique graph''', '''representing graph''', '''primal graph''', '''Gaifman graph''') of a hypergraph is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge.