Editing Hypergraph (section)

== Related hypergraphs ==
Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called ''subhypergraphs'', ''partial hypergraphs'' and ''section hypergraphs''.

Let <math>H=(X,E)</math> be the hypergraph consisting of vertices

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

and having ''edge set''

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

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

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 \mid e \in E, 
e \cap A \neq \emptyset \rbrace \right).</math>

An alternative term is the '''restriction of ''H'' to ''A'''''.<ref name="lp">{{Cite Lovasz Plummer}}</ref>{{rp|468}}

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> is fully contained in the extension <math>Ex(H_A)</math>. Formally
:<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 \mid e \subseteq (A \cup A') \rbrace</math>.

The '''partial hypergraph''' is a hypergraph with some edges removed.<ref name="lp" />{{rp|468}} 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

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

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

:<math>H \times A = \left(A, \lbrace e_i \mid 
i\in I_e, 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

:<math>X_m = \lbrace e_i \mid 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.,

:<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>''.

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.