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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}，
和边集
𝐸={𝑒𝑖|𝑖∈𝐼𝑒∧𝑒𝑖⊆𝑋∧𝑒𝑖≠∅𝐸}
其中 𝐼𝑣 和 𝐼𝑒 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
:<math>H_A=\left(A, \lbrace e \cap A | e \in E \land
:<math>H_A=\left(A, \lbrace e \cap A | e \in E \land
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：
𝐻𝐴=(𝐴,{𝑒𝐴∩∩|𝑒𝐴∈𝐸∧𝑒∩𝐴≠∅)
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：
:<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
:<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
:<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>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：
:<math>\left(H^*\right)^* = H.</math>
:<math>\left(H^*\right)^* = H.</math>
| last5 = Verbeek | first5 = Kevin
| last5 = Verbeek | first5 = Kevin
| contribution = On planar supports for hypergraphs
| contribution = On planar supports for hypergraphs
| doi = 10.1007/978-3-642-11805-0_33
| doi = 10.1007/_33
| pages = 345–356
| pages = 345–356
| publisher = Springer-Verlag
| publisher = Springer-Verlag
| volume = 5849
| volume = 5849
| year = 2010| title-link = International Symposium on Graph Drawing
| year = 2010| title-link = International Symposium on Graph Drawing
| isbn = 978-3-642-11804-3
| isbn =
| doi-access = free
| doi-access = free
}}.</ref>
}}.</ref>