更改

where <math>I_v</math> and <math>I_e</math> are the [[index set]]s of the vertices and edges respectively.

where <math>I_v</math> and <math>I_e</math> are the [[index set]]s of the vertices and edges respectively.

让 𝐻=(𝑋,𝐸) 是一个超图，包含顶点集：

 

𝑋={𝑥𝑖|𝑖∈𝐼𝑣}，

让 <math>H=(X,E)</math> 是一个超图，包含顶点集：

 

𝐸={𝑒𝑖|𝑖∈𝐼𝑒∧𝑒𝑖⊆𝑋∧𝑒𝑖≠∅𝐸}

:<math>X = \lbrace x_i | i \in I_v \rbrace,</math>

其中 𝐼𝑣 和 𝐼𝑒 分别是顶点和边集的索引集。

 

 

:<math>E = \lbrace e_i | i\in I_e \land e_i \subseteq X \land e_i \neq \emptyset \rbrace,</math>

 

其中 <math>I_v</math> 和 <math>I_e</math> 分别是顶点和边集的[[索引集]]

A ''subhypergraph'' is a hypergraph with some vertices removed.  Formally, the subhypergraph <math>H_A</math> induced by <math>A \subseteq X </math> is defined as

A ''subhypergraph'' is a hypergraph with some vertices removed.  Formally, the subhypergraph <math>H_A</math> induced by <math>A \subseteq X </math> is defined as

e \cap A \neq \emptyset \rbrace \right).</math>

e \cap A \neq \emptyset \rbrace \right).</math>

子超图是去掉某些顶点的超图。在形式上，若 𝐴⊆𝑋 是顶点子集，则子超图 𝐻𝐴 被定义为：

''子超图'' 是去掉某些顶点的超图。在形式上，若 <math>A \subseteq X </math> 是顶点子集，则子超图 <math>H_A</math> 被定义为：

𝐻𝐴=(𝐴,{𝑒𝐴∩∩|𝑒𝐴∈𝐸∧𝑒∩𝐴≠∅)

 

:<math>H_A=\left(A, \lbrace e \cap A | e \in E \land

e \cap A \neq \emptyset \rbrace \right).</math>

An ''extension'' of a ''subhypergraph'' is a hypergraph where each

An ''extension'' of a ''subhypergraph'' is a hypergraph where each

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 H 的超边都部分包含在子超图的 𝐻𝐴，并且完全包含在扩展的𝐸𝑥(𝐻𝐴) 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> 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>.

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 𝐽⊂𝐼𝑒 ，由 𝐽 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

 

:<math>\left(X, \lbrace e_i | i\in J \rbrace \right).</math>

:<math>\left(X, \lbrace e_i | i\in J \rbrace \right).</math>

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 𝐴⊆𝑋，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 |  

i\in I_e \land e_i \subseteq A  \rbrace \right).</math>

i\in I_e \land e_i \subseteq A  \rbrace \right).</math>

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

𝐻 的重记号 𝐻∗ 则是一个顶点和边互换的超图，因此顶点由 {𝑒𝑖 } 给出，边由 {𝑋𝑚} 给出，其中

<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>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>