[permalink] [id link]

A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BG, and ( x < sub > 1 </ sub >, e < sub > 1 </ sub >) are connected with an edge if and only if vertex x < sub > 1 </ sub > is contained in edge e < sub > 1 </ sub > in H. Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above.

from
Wikipedia

## Some Related Sentences

hypergraph and H

**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__induces

**a**

**connected**subgraph

**in**G

**.**For

**a**disconnected

__hypergraph__

__H__

**,**G

**is**

**a**host

**graph**

**if**there

**is**

**a**bijection between

**the**

**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 set

**of**automorphisms**of****a**__hypergraph____H__(=**(****X****,****E**))**is****a**group under composition**,**called**the**automorphism group**of****the**__hypergraph__**and**written Aut**(**__H__).**A**transversal

**(**or " hitting set ")

**of**

**a**

__hypergraph__

__H__=

**(**

**X**

**,**

**E**)

**is**

**a**set that has nonempty intersection

**with**every

**edge**

**.**

The transversal

__hypergraph__**of**__H__**is****the**__hypergraph__**(****X****,**F ) whose**edge**set F consists**of**all minimal transversals**of**__H__**.**

hypergraph and may

In computational geometry

**,****a**__hypergraph____may__sometimes**be**called**a**range space**and**then**the**hyperedges**are**called ranges**.**
The hyperedges

**of****the**__hypergraph__**are****represented****by**contiguous subsets**of**these regions**,**which__may__**be**indicated**by**coloring**,****by**drawing outlines around them**,**or both**.**
An order-n Venn diagram

**,**for instance**,**__may__**be**viewed**as****a**subdivision drawing**of****a**__hypergraph__**with**n hyperedges**(****the**curves defining**the**diagram )**and**2**<**sup**>**n**</**sup**>**−**1**vertices**(****represented****by****the**regions into which these curves subdivide**the**plane ).
In contrast

**with****the**polynomial-time recognition**of**planar graphs**,**it**is**NP-complete to determine whether**a**__hypergraph__has**a**planar subdivision drawing**,**but**the**existence**of****a**drawing**of**this type__may__**be**tested efficiently when**the**adjacency pattern**of****the**regions**is**constrained to**be****a**path**,**cycle**,**or tree**.****A**

__hypergraph__

**is**

**a**combinatorial structure that

**,**like

**an**undirected

**graph**

**,**has vertices

**and**edges

**,**but

**in**which

**the**edges

__may__

**be**arbitrary

**sets**

**of**vertices rather than having to have exactly two endpoints

**.**

**A**

**bipartite**

**graph**

__may__

**be**used to model

**a**

__hypergraph__

**in**which

**is**

**the**set

**of**vertices

**of**

**the**

__hypergraph__

**,**

**is**

**the**set

**of**hyperedges

**,**

**and**contains

**an**

**edge**from

**a**

__hypergraph__

**vertex**to

**a**

__hypergraph__

**edge**exactly when

**is**one

**of**

**the**endpoints

**of**

**.**

As

**a**special case**of**this correspondence between**bipartite**graphs**and**hypergraphs**,****any**multigraph**(****a****graph****in**which there__may__**be**two or more edges between**the**same two vertices )__may__**be**interpreted**as****a**__hypergraph__**in**which**some**hyperedges have equal**sets****of**endpoints**,****and****represented****by****a****bipartite****graph**that does not have multiple adjacencies**and****in**which**the**vertices on one side**of****the**bipartition all have degree two**.****(**Since

**a**family

**of**

**sets**

__may__

**be**called

**a**

__hypergraph__

**,**

**and**since every set

**in**has size r

**,**

**is**

**a**uniform

__hypergraph__

**of**rank r .)

The edges

**of****a**__hypergraph____may__form**an**arbitrary family**of****sets****,**so**the**line**graph****of****a**__hypergraph__**is****the**same**as****the**intersection**graph****of****the****sets**from**the**family**.**
The cliques

**of**size**in****the**Erdős – Faber – Lovász conjecture__may__**be**interpreted**as****the**hyperedges**of**an-uniform linear__hypergraph__that has**the**same vertices**as****the**underlying**graph****.**

hypergraph and be

Most classes

**of**CSPs that**are**known to__be__tractable**are**those where**the**__hypergraph__**of**constraints has bounded treewidth**(****and**there**are****no**restrictions on**the**set**of**constraint relations ), or where**the**constraints have arbitrary form but there exist essentially non-unary polymorphisms**of****the**set**of**constraint relations**.**
In particular

**,**there**is****a****bipartite**" incidence**graph**" or " Levi**graph**" corresponding to every__hypergraph__**,****and**conversely**,**most**,**but not all**,****bipartite**graphs can__be__regarded**as**incidence graphs**of**hypergraphs**.****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 contains at least one

**vertex**from both classes

**.**

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**

__hypergraph__

**is**said to

__be__vertex-transitive

**(**or vertex-symmetric )

**if**all

**of**its vertices

**are**symmetric

**.**

This circuit diagram can

__be__interpreted**as****a**drawing**of****a**__hypergraph__**in**which four vertices**(**depicted**as**white rectangles**and**disks )**are****connected****by**three hyperedges drawn**as**trees**.**
An order-4 Venn diagram

**,**which can__be__interpreted**as****a**subdivision drawing**of****a**__hypergraph__**with**15 vertices**(****the**15 colored regions )**and**4 hyperedges**(****the**4 ellipses ).

hypergraph and represented

In

**the**UNL approach**,**information conveyed**by**natural language**is**__represented__**,**sentence**by**sentence**,****as****a**__hypergraph__composed**of****a**set**of**directed binary labeled links**(**referred to**as**relations ) between**nodes**or hypernodes**(****the**Universal Words**,**or simply UW ), which stand for concepts**.**

hypergraph and by

The dual

**of****is****a**__hypergraph__whose vertices**and**edges**are**interchanged**,**so that**the**vertices**are**given__by__**and**whose edges**are**given__by__where

hypergraph and graph

In mathematics

**,****a**__hypergraph__**is****a**generalization**of****a**__graph__**in**which**an****edge**can connect**any**number**of**vertices**.**
However

**,**it**is**often desirable to study hypergraphs where all hyperedges have**the**same cardinality ;**a**k-uniform__hypergraph__**is****a**__hypergraph__such that all its hyperedges have size k**.****(**In other words**,**it**is****a**collection**of****sets****of**size k .) So**a**2-uniform__hypergraph__**is****a**__graph__**,****a**3-uniform__hypergraph__**is****a**collection**of**unordered triples**,****and**so on**.**
The primal

__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**.**

hypergraph and sets

The intersection number

**of****a****graph****is****the**minimum number**of**elements**in****a**family**of**__sets__whose intersection**graph****is****,**or equivalently**the**minimum number**of**vertices**in****a**__hypergraph__whose line**graph****is****.**

hypergraph and E

**A**partition theorem due to

__E__

**.**Dauber states that

**,**for

**an**edge-transitive

__hypergraph__

**,**there exists

**a**partition

Equivalently

**,****a**Sperner family**is****an**antichain**in****the**inclusion lattice over**the**power set**of**__E__**.****A**Sperner family**is**also sometimes called**an**independent system or**,****if**viewed from**the**__hypergraph__perspective**,****a**clutter**.**

hypergraph and are

Because

__hypergraph__links can have**any**cardinality**,**there__are__several notions**of****the**concept**of****a**subgraph**,**called subhypergraphs**,**partial hypergraphs**and**section hypergraphs**.**
When

**the**edges**of****a**__hypergraph____are__explicitly labeled**,**one has**the**additional notion**of**strong isomorphism**.**
When

**the**vertices**of****a**__hypergraph____are__explicitly labeled**,**one has**the**notions**of**equivalence**,****and**also**of**equality**.****A**simple

__hypergraph__

**is**

**a**

__hypergraph__

**in**which at most one hyperedge connects

**any**pair

**of**vertices

**and**there

__are__

**no**hyperedges

**of**size at most one

**.**

0.100 seconds.