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Formally, a vertex cover of a graph G is a set C of vertices such that each edge of G is incident to at least one vertex in C. The set C is said to cover the edges of G. The following figure shows examples of vertex covers in two graphs ( and the set C is marked with red ).

from
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## Some Related Sentences

Formally and vertex

__Formally__

**,**given

**a**

**graph**

**G**

**,**

**a**

__vertex__labeling

**is**

**a**function mapping

**vertices**

**of**

**G**

**to**

**a**

**set**

**of**labels

**.**

Formally and cover

__Formally__

**,**

**a**Markov partition

**is**used

**to**provide

**a**finite

__cover__for

**the**smooth system ;

**each**

**set**

**of**

**the**

__cover__

**is**associated

**with**

**a**single symbol

**,**

**and**

**the**sequences

**of**symbols result as

**a**trajectory

**of**

**the**system moves from

**one**

**of**

**the**covering sets

**to**another

**.**

Formally and graph

__Formally__

**,**

**a**

__graph__rewriting system usually consists

**of**

**a**

**set**

**of**

__graph__rewrite rules

**of**

**the**form

**,**

**with**being called pattern

__graph__

**(**or left-hand side )

**and**being called replacement

__graph__

**(**or right-hand side

**of**

**the**rule

**).**

__Formally__

**,**

**a**

__graph__

**is**symmetric

**with**respect

**to**

**the**origin if it

**is**unchanged when reflected across both

**the**x-axis

**and**y-axis

**.**

__Formally__

**,**

**a**signed

__graph__Σ

**is**

**a**pair

**(**

**G**

**,**σ )

**that**consists

**of**

**a**

__graph__

**G**=

**(**V

**,**E )

**and**

**a**sign mapping or signature σ from E

**to**

**the**sign group

__Formally__

**,**

**a**biased

__graph__Ω

**is**

**a**pair

**(**

**G**

**,**B ) where B

**is**

**a**linear class

**of**circles ; this by definition

**is**

**a**class

**of**circles

**that**satisfies

**the**theta-graph property mentioned above

**.**

__Formally__

**,**

**the**upper density

**of**

**a**

__graph__

**G**

**is**

**the**infimum

**of**

**the**values α

**such**

**that**

**the**finite subgraphs

**of**

**G**

**with**density α have

**a**bounded number

**of**

**vertices**

**.**

Formally and G

Informally

**,**__G__has**the**above presentation if it**is****the**" freest group " generated by S subject only**to****the**relations R**.**__Formally__**,****the**group__G__**is****said****to**have**the**above presentation if it**is**isomorphic**to****the**quotient**of****a**free group on S by**the**normal subgroup generated by**the**relations R**.**__Formally__

**,**

**the**index

**of**H

**in**

__G__

**is**defined as

**the**number

**of**cosets

**of**H

**in**

__G__

**.**

**(**

**The**number

**of**left cosets

**of**H

**in**

__G__

**is**always equal

**to**

**the**number

**of**right cosets

**.**

__Formally__

**,**given

**two**categories

**C**

**and**D

**,**an equivalence

**of**categories consists

**of**

**a**functor F:

**C**→ D

**,**

**a**functor

__G__: D →

**C**

**,**

**and**

**two**natural isomorphisms ε: FG → I < sub > D </ sub >

**and**η: I < sub >

**C**</ sub >→ GF

**.**

__Formally__

**,**

**a**frame on

**a**homogeneous space

__G__/ H consists

**of**

**a**point

**in**

**the**tautological bundle

__G__→

__G__/ H

**.**

__Formally__

**,**let

__G__be

**a**Coxeter group

**with**reduced root system R

**and**k < sub > v </ sub >

**a**multiplicity function on R

**(**so k < sub > u </ sub > = k < sub > v </ sub > whenever

**the**reflections σ < sub > u </ sub >

**and**σ < sub > v </ sub > corresponding

**to**

**the**roots u

**and**v are conjugate

**in**

__G__

**).**

Formally and is

__Formally__organized vocational programs supported by federal funds allow high school students

**to**gain experience

**in**

**a**field

**of**work which

__is__likely

**to**lead

**to**

**a**full-time job on graduation

**.**

__Formally__

**,**

**the**

**set**

**of**all context-free languages

__is__identical

**to**

**the**

**set**

**of**languages accepted by pushdown automata

**(**PDA

**).**

More rigorously

**,****the**divergence**of****a**vector field F**at****a**point p__is__defined as**the**limit**of****the**net flow**of**F across**the**smooth boundary**of****a**three dimensional region V divided by**the**volume**of**V as V shrinks**to**p**.**__Formally__**,**__Formally__

**,**

**the**base

__is__known as Naval Support Facility Diego Garcia

**(**

**the**US activity ) or Permanent Joint Operating Base

**(**PJOB ) Diego Garcia

**(**

**the**UK's term

**).**

__Formally__

**,**there

__is__

**a**clear distinction: " DFT " refers

**to**

**a**mathematical transformation or function

**,**regardless

**of**how it

__is__computed

**,**whereas " FFT " refers

**to**

**a**specific family

**of**algorithms for computing DFTs

**.**

__Formally__

**,**oxidation state

__is__

**the**hypothetical charge

**that**an atom would have if all bonds

**to**atoms

**of**different elements were 100 % ionic

**.**

__Formally__

**,**an inner product space

__is__

**a**vector space V over

**the**field together

**with**an inner product

**,**i

**.**e.,

**with**

**a**map

__Formally__

**,**if M

__is__

**a**

**set**

**,**

**the**identity function f on M

__is__defined

**to**be

**that**function

**with**domain

**and**codomain M which satisfies

*

__Formally__**,**when working over**the**reals**,**as here**,**this__is__accomplished by considering**the**limit as ε → 0 ; but**the**" infinitesimal " language generalizes directly**to**Lie groups over general rings**.**__Formally__

**,**

**a**profinite group

__is__

**a**Hausdorff

**,**compact

**,**

**and**totally disconnected topological group:

**that**

__is__

**,**

**a**topological group

**that**

__is__also

**a**Stone space

**.**

__Formally__

**,**this sharing

**of**dynamics

__is__referred

**to**as universality

**,**

**and**systems

**with**precisely

**the**same critical exponents are

**said**

**to**belong

**to**

**the**same universality class

**.**

__Formally__

**,**

**a**frame

__is__defined

**to**be

**a**lattice L

**in**which finite meets distribute over arbitrary joins

**,**i

**.**e

**.**every

**(**even infinite ) subset

Formally and set

__Formally__speaking

**,**

**a**collation method typically defines

**a**total order on

**a**

__set__

**of**possible identifiers

**,**called sort keys

**,**which consequently produces

**a**total preorder on

**the**

__set__

**of**items

**of**information

**(**items

**with**

**the**same identifier are not placed

**in**any defined order

**).**

__Formally__

**,**according

**to**

**the**Constitution

**,**citizens

**of**Turkmenistan have

**the**right

**to**

__set__up political parties

**and**other public associations

**,**acting within

**the**framework

**of**

**the**Constitution

**and**laws

**,**

**and**public associations

**and**groups

**of**citizens have

**the**right

**to**nominate their candidates

**in**accordance

**with**

**the**election law

**.**

__Formally__

**,**

**the**convex hull may be defined as

**the**intersection

**of**all convex sets containing X or as

**the**

__set__

**of**all convex combinations

**of**points

**in**X

**.**

__Formally__

**,**

**the**discrete cosine transform

**is**

**a**linear

**,**invertible function

**(**where denotes

**the**

__set__

**of**real numbers ), or equivalently an invertible N × N square matrix

**.**

__Formally__

**,**we are given

**a**

__set__

**of**hypotheses

**and**

**a**

__set__

**of**manifestations ; they are related by

**the**domain knowledge

**,**represented by

**a**function

**that**takes as an argument

**a**

__set__

**of**hypotheses

**and**gives as

**a**result

**the**corresponding

__set__

**of**manifestations

**.**

__Formally__

**,**

**a**function ƒ

**is**real analytic on an open

__set__D

**in**

**the**real line if for any x < sub > 0 </ sub >

**in**D

**one**can write

__Formally__

**,**

**the**movement

**is**

**a**rondo

**that**acts as

**the**theme for

**a**

__set__

**of**eight variations

**,**capped off by

**a**dramatic coda

**.**

__Formally__

**,**

**the**discrete sine transform

**is**

**a**linear

**,**invertible function F: R < sup > N </ sup > < tt >-></ tt > R < sup > N </ sup >

**(**where R denotes

**the**

__set__

**of**real numbers ), or equivalently an N × N square matrix

**.**

__Formally__

**,**

**the**discrete Hartley transform

**is**

**a**linear

**,**invertible function H: R < sup > n </ sup > < tt >-></ tt > R < sup > n </ sup >

**(**where R denotes

**the**

__set__

**of**real numbers

**).**

__Formally__

**,**

**a**constraint satisfaction problem

**is**defined as

**a**triple

**,**where

**is**

**a**

__set__

**of**variables

**,**

**is**

**a**domain

**of**values

**,**

**and**

**is**

**a**

__set__

**of**constraints

**.**

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