aarjaneiro/planetmath huggingface dataset:
Hasse diagram

Hasse diagram

If

(

A

,

≤

)

is a finite poset, then it can be represented by a

Hasse diagram

, which is a graph whose vertices are elements of

A

and the edges correspond to the covering relation. More precisely an edge from

x

∈

A

to

y

∈

A

is present if

•

x

<

y

.

•

There is no

z

∈

A

such that

x

<

z

and

z

<

y

. (There are no in-between

elements

.)

If

x

<

y

, then in

y

is drawn higher than

x

. Because of that, the direction of the edges is never indicated in a Hasse diagram.

Example:

If

A

=

𝒫

⁢

(

{

1

,

2

,

3

}

)

, the

power set

of

{

1

,

2

,

3

}

, and

≤

is the subset

relation

⊆

, then Hasse diagram is

\xymatrix

⁢

&

⁢

{

1

,

2

,

3

}

⁢

&

⁢

{

1

,

2

}

⁢

\ar

⁢

@

-

[

u

⁢

r

]

⁢

&

⁢

{

1

,

3

}

⁢

\ar

⁢

@

-

[

u

]

⁢

&

⁢

{

2

,

3

}

⁢

\ar

⁢

@

-

[

u

⁢

l

]

⁢

{

1

}

⁢

\ar

⁢

@

-

[

u

]

⁢

\ar

⁢

@

-

[

u

⁢

r

]

⁢

&

⁢

{

2

}

⁢

\ar

⁢

@

-

[

u

⁢

l

]

⁢

\ar

⁢

@

-

[

u

⁢

r

]

⁢

&

⁢

{

3

}

⁢

\ar

⁢

@

-

[

u

⁢

l

]

⁢

\ar

⁢

@

-

[

u

]

⁢

&

⁢

∅

⁢

\ar

⁢

@

-

[

u

⁢

l

]

⁢

\ar

⁢

@

-

[

u

]

⁢

\ar

⁢

@

-

[

u

⁢

r

]

⁢

&

Even though

{

3

}

<

{

1

,

2

,

3

}

(since

{

3

}

⊂

{

1

,

2

,

3

}

), there is no edge directly between them because there are inbetween elements:

{

2

,

3

}

and

{

1

,

3

}

. However, there still remains an indirect path from

{

3

}

to

{

1

,

2

,

3

}

.

Title

Hasse diagram

Canonical name

HasseDiagram

Date of creation

2013-03-22 12:15:23

Last modified on

2013-03-22 12:15:23

Owner

bbukh (348)

Last modified by

bbukh (348)

Numerical id

18

Author

bbukh (348)

Entry type

Definition

Classification

msc 05C90

Related topic

Poset

Related topic

PartialOrder

Generated on Fri Feb 9 07:56:51 2018 by

LaTeXML
block

block

A

subgraph

B

of a graph

G

is a

block of

G

if either it is a bridge (together with the vertices

incident

with the bridge) or else it is a maximal 2-connected subgraph of

G

.

Any two blocks of a graph

G

have at most one vertex in common. Also, every vertex belonging to at least two blocks is a

cutvertex

of

G

, and, conversely, every cutvertex belongs to at least two blocks.

Adapted with permission of the author from

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

block

Canonical

name

Block

Date of creation

2013-03-22 12:32:00

Last modified on

2013-03-22 12:32:00

Owner

digitalis (76)

Last modified by

digitalis (76)

Numerical id

4

Author

digitalis (76)

Entry type

Definition

Classification

msc 05C99

Related topic

Cutvertex

Related topic

Bridge

Generated on Fri Feb 9 07:56:53 2018 by

LaTeXML
bridge

bridge

A

bridge

of a graph

G

is an edge whose deletion increases the number of

components

of

G

. The vertex analogue of a bridge is a

cutvertex

.

Title

bridge

Canonical

name

Bridge

Date of creation

2013-03-22 12:31:41

Last modified on

2013-03-22 12:31:41

Owner

digitalis (76)

Last modified by

digitalis (76)

Numerical id

5

Author

digitalis (76)

Entry type

Definition

Classification

msc 05C99

Related topic

Cutvertex

Related topic

ConnectedGraph

Related topic

Block

Generated on Fri Feb 9 07:56:55 2018 by

LaTeXML
complete graph

complete graph

The

complete graph

with

n

vertices, denoted

K

n

, contains all possible edges; that is, any two vertices are

adjacent

.

The complete graph of

4

vertices, or

K

4

looks like this:

The number of edges in

K

n

is the

n

-

1

th

triangular number

. Every vertex in

K

n

has degree

n

-

1

; therefore

K

n

has an

Euler circuit

if and only if

n

is odd. A complete graph always has a

Hamiltonian path

, and the

chromatic number

of

K

n

is always

n

.

Title

complete graph

Canonical name

CompleteGraph

Date of creation

2013-03-22 12:16:57

Last modified on

2013-03-22 12:16:57

Owner

vampyr (22)

Last modified by

vampyr (22)

Numerical id

10

Author

vampyr (22)

Entry type

Definition

Classification

msc 05C99

Synonym

complete

Synonym

clique

Related topic

Tournament

Generated on Fri Feb 9 07:56:56 2018 by

LaTeXML
edge-contraction

edge-contraction

Given an edge

x

⁢

y

of a graph

G

, the graph

G

/

x

⁢

y

is obtained from

G

by

contracting

the edge

x

⁢

y

; that is, to get

G

/

x

⁢

y

we identify the vertices

x

and

y

and remove all loops and duplicate edges. A graph

G

′

obtained by a

sequence

of edge-contractions is said to be a

contraction

of

G

.

Adapted with permission of the author from

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

edge-contraction

Canonical name

Edgecontraction

Date of creation

2013-03-22 12:31:43

Last modified on

2013-03-22 12:31:43

Owner

rspuzio (6075)

Last modified by

rspuzio (6075)

Numerical id

5

Author

rspuzio (6075)

Entry type

Definition

Classification

msc 05C99

Related topic

TheoremOn3ConnectedGraphs

Defines

contraction

Generated on Fri Feb 9 07:56:58 2018 by

LaTeXML
Euler’s polyhedron theorem

Euler’s polyhedron theorem

Theorem 1

If a

connected

plane graph

has

n

vertices,

m

edges, and

f

faces, then

n

-

m

+

f

=

2

.

Title

Euler’s

polyhedron

theorem

Canonical name

EulersPolyhedronTheorem

Date of creation

2013-03-22 12:25:28

Last modified on

2013-03-22 12:25:28

Owner

Mathprof (13753)

Last modified by

Mathprof (13753)

Numerical id

6

Author

Mathprof (13753)

Entry type

Theorem

Classification

msc 05C99

Synonym

Euler’s formula

Related topic

Graph

Related topic

GraphTheory

Related topic

PoincareFormula

Related topic

Polytope

Generated on Fri Feb 9 07:57:00 2018 by

LaTeXML
Euler’s polyhedron theorem, proof of

Euler’s polyhedron theorem, proof of

This proof is not one of the standard proofs given to Euler’s

formula

. I found the idea presented in one of Coxeter’s books. It presents a different approach to the formula, that may be more familiar to modern students who have been exposed to a “

Discrete Mathematics

” course. It falls into the category of “informal” proofs: proofs which assume without proof certain properties of

planar graphs

usually proved with algebraic topology. This one makes deep (but somewhat hidden) use of the

Jordan curve theorem

.

Let

G

=

(

V

,

E

)

be a planar graph; we consider some particular

planar embedding

of

G

. Let

F

be the set of

faces

of this

embedding

. Also let

G

′

=

(

F

,

E

′

)

be the

dual graph

(

E

′

contains an edge between any 2

adjacent

faces of

G

). The planar embeddings of

G

and

G

′

determine a correspondence between

E

and

E

′

: two vertices of

G

are adjacent iff they both belong to a pair of adjacent faces of

G

; denote by

ϕ

:

E

→

E

′

this correspondence.

In all illustrations, we represent a planar graph

G

, and the two sets of edges

T

⊆

E

(in red) and

T

′

⊆

E

′

(in blue).

Let

T

⊆

E

be a

spanning tree

of

G

. Let

T

′

=

E

′

∖

ϕ

⁢

[

E

]

. We claim that

T

′

is a spanning tree of

G

′

. Indeed,

T

′

contains no loop.

Given any loop of edges in

T

′

, we may draw a loop on the faces of

G

which participate in the loop. This loop must

partition

the vertices of

G

into two non-empty sets, and only crosses edges of

E

∖

T

. Thus,

(

V

,

T

)

has more than a single

connected component

, so

T

is not

spanning

.

[The proof of this utilizes the Jordan curve theorem.]

T

′

spans

G

′

.

For suppose

T

′

does not connect all faces

F

. Let

f

1

,

f

2

∈

F

be two faces with no path between them in

T

′

. Then

T

must contain a cycle

separating

f

1

from

f

2

, and cannot be a tree.

[The proof of this utilizes the Jordan curve theorem.]

We thus have a partition

E

=

T

∪

ϕ

-

1

⁢

[

T

′

]

of the edges of

G

into two sets. Recall that in any tree, the number of edges is one less than the number of vertices. It follows that

|

E

|

=

|

T

|

+

|

T

′

|

=

(

|

V

|

-

1

)

+

(

|

F

|

-

1

)

=

|

V

|

+

|

F

|

-

2

,

as required.

Title

Euler’s polyhedron

theorem

, proof of

Canonical name

EulersPolyhedronTheoremProofOf

Date of creation

2013-03-22 12:47:36

Last modified on

2013-03-22 12:47:36

Owner

mps (409)

Last modified by

mps (409)

Numerical id

7

Author

mps (409)

Entry type

Proof

Classification

msc 05C99

Generated on Fri Feb 9 07:57:02 2018 by

LaTeXML
graph

graph

A

graph

G

is an

ordered pair

of

disjoint sets

(

V

,

E

)

such that

E

is a subset of the set

V

(

2

)

of unordered pairs of

V

.

V

and

E

are always assumed to be finite, unless explicitly stated otherwise. The set

V

is the set of

vertices

(sometimes called nodes) and

E

is the set of

edges

. If

G

is a graph, then

V

=

V

⁢

(

G

)

is the

vertex set

of

G

, and

E

=

E

⁢

(

G

)

is the

edge set

.
Typically,

V

⁢

(

G

)

is defined to be nonempty. If

x

is a vertex of

G

, we sometimes write

x

∈

G

instead of

x

∈

V

⁢

(

G

)

.

An edge

{

x

,

y

}

(with

x

≠

y

) is said to

join

the vertices

x

and

y

and is denoted by

x

⁢

y

. One says that the edges

x

⁢

y

and

y

⁢

x

are

equivalent

; the vertices

x

and

y

are the

endvertices

of this edge. If

x

⁢

y

∈

E

⁢

(

G

)

, then

x

and

y

are

adjacent

, or

neighboring

, vertices of

G

, and the vertices

x

and

y

are

incident

with the edge

x

⁢

y

. Two edges are

adjacent

if they have at least one common endvertex. Also,

x

∼

y

means that the vertex

x

is adjacent to the vertex

y

.

Notice that this

definition

allows pairs of the form

{

x

,

x

}

, which would correspond to a node joining to itself. Some authors explicitly disallow this in their definition of a graph.

Some graphs.

Note: Some authors include

multigraphs

in their definition of a graph. In this notation, the above definition corresponds to that of a

simple graph

. A graph is then

simple

if there is at most one edge joining each pair of nodes.

Adapted with permission of the author from

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

graph

Canonical

name

Graph

Date of creation

2013-03-22 11:57:54

Last modified on

2013-03-22 11:57:54

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

34

Author

mathcam (2727)

Entry type

Definition

Classification

msc 05C99

Related topic

LoopOfAGraph

Related topic

NeighborhoodOfAVertex

Related topic

EulersPolyhedronTheorem

Related topic

Digraph

Related topic

Tree

Related topic

SpanningTree

Related topic

ConnectedGraph

Related topic

Cycle

Related topic

GraphTheory

Related topic

GraphTopology

Related topic

Subgraph

Related topic

SimplePath

Related topic

EulerPath

Related topic

Diameter3

Related topic

DistanceInAGraph

Related topic

GraphHomomorphism

Related topic

Pseudograph

Related topic

Multigraph

Related topic

OrderOf

Defines

edge

Defines

vertex

Defines

endvertex

Defines

adjacent

Defines

incident

Defines

join

Defines

vertices

Defines

simple graph

Generated on Fri Feb 9 07:57:05 2018 by

LaTeXML
graph homeomorphism

graph homeomorphism

Let

G

=

(

V

,

E

)

be a simple

undirected graph

.
A

simple subdivision

is the replacement of an edge

(

x

,

y

)

∈

E

with a pair of edges

(

x

,

z

)

,

(

z

,

y

)

,

z

being a new vertex,

i.e.

,

z

∉

V

.
The reverse

operation

of a simple subdivision
is an edge-contraction through a vertex of degree 2.

Two graphs

G

1

,

G

2

are

homeomorphic

if

G

1

can be transformed into

G

2

via a

finite sequence

of simple subdivisions
and edge-contractions through vertices of degree 2.
It is easy to see that graph homeomorphism is an

equivalence relation

.

Equivalently,

G

1

and

G

2

are homeomorphic
if there exists a third graph

G

3

such that both

G

1

and

G

2

can be obtained from

G

3

via a finite sequence of edge-contractions through vertices of degree 2.

If a graph

G

has a

subgraph

H

which is homeomorphic to a graph

G

′

having no vertices of degree 2,
then

G

′

is a minor of

G

.
The vice versa is not true:
as a counterexample, the

Petersen graph

has

K

5

as a minor, but no subgraph homeomorphic to

K

5

.
This happens because a graph homeomorphism
cannot change the number of vertices of degree

d

≠

2

:
since all the vertices of

K

5

have degree 4
and all the vertices of the Petersen graph have degree 3,
no subgraph of the Petersen graph can be homeomorphic to

K

5

.

Title

graph homeomorphism

Canonical name

GraphHomeomorphism

Date of creation

2013-03-22 18:02:02

Last modified on

2013-03-22 18:02:02

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

8

Author

Ziosilvio (18733)

Entry type

Definition

Classification

msc 05C99

Defines

simple subdivision

Generated on Fri Feb 9 07:57:07 2018 by

LaTeXML
graph minor theorem

graph minor theorem

If

(

G

k

)

k

∈

ℕ

is an

infinite

sequence

of

finite graphs

, then there exist two numbers

m

<

n

such that

G

m

is

isomorphic

to a minor of

G

n

.

This

theorem

is often referred to as the deepest result in

graph theory

. It was proven by Robertson and Seymour in 1988 as the culminating result of a long series of articles, and the proof was recently published in the Journal of Combinatorial Theory series B. It resolves Wagner’s

conjecture

in the affirmative and leads to an important

generalization

of Kuratowski’s theorem.

Specifically, to every set

𝒢

of finite graphs which is

closed under

taking minors (meaning if

G

∈

𝒢

and

H

is isomorphic to a minor of

G

, then

H

∈

𝒢

) there exist

finitely many

graphs

G

1

,

…

,

G

n

such that

𝒢

consists precisely of those finite graphs that do not have a minor isomorphic to one of the

G

i

. The graphs

G

1

,

…

,

G

n

are often referred to as the

forbidden minors

for the class

𝒢

.

0.1

References

•

Neil Robertson and P.D. Seymour.

Graph Minors

XX: Wagner’s conjecture.

Journal of Combinatorial Theory B

, Volume 92, Issue 2, November 2004, Pages 325-357.

\url

http://dx.doi.org/10.1016/j.jctb.2004.08.001

Title

graph minor theorem

Canonical

name

GraphMinorTheorem

Date of creation

2013-03-22 14:04:20

Last modified on

2013-03-22 14:04:20

Owner

AxelBoldt (56)

Last modified by

AxelBoldt (56)

Numerical id

16

Author

AxelBoldt (56)

Entry type

Theorem

Classification

msc 05C99

Related topic

MinorOfAGraph

Generated on Fri Feb 9 07:57:08 2018 by

LaTeXML
handshake lemma

handshake lemma

We recall that the degree (sometimes called

valency

)
of a vertex

v

of an

undirected graph

G

is the number

δ

⁢

(

v

)

of

v

’s neighbors,

i.e.

, the number of vertices

z

of

G

such that an edge

(

v

,

z

)

exists.

Theorem 1

(Handshake lemma)

Let

G

=

(

V

,

E

)

be a simple nondirected

finite graph

. Then

∑

v

∈

V

δ

⁢

(

v

)

=

2

⁢

|

E

|

.

(1)

Proof.

List all the edges of

G

, with their

endpoints

, each edge once.
Each vertex

v

occurs a number of times
equal to the number of edges

v

is an endpoint of,

i.e.

, the number

δ

⁢

(

v

)

of its neighbors.
On the other hand, each edge has two endpoints,
thus the sum of all

occurrences

of vertices
must equal twice the number of edges.

□

Theorem 2

A simple finite undirected graph
has an

even number

of vertices of odd degree.

Proof.

By the

handshake lemma

,
the sum of the degrees of all vertices of odd degree must be even,
which is only possible if their number is even.

□

The following two statements about trees also follow from the handshake lemma.
Recall that the

maximum degree

of a node in a finite tree
cannot exceed the number of leaves of the tree,
because each

subtree

of any node contains at least one leaf.

Theorem 3

A finite tree with exactly three leaves
is

homeomorphic

to

K

1

,

3

.

Proof.

A finite tree with three leaves can have no vertex of degree greater than 3.
By the handshake lemma, the number of vertices of odd degree must be even:
this forces a vertex of degree 3 to exist.
Such a vertex is also unique,
because if there were two, the tree would have at least four leaves.
All the other vertices, except the leaves, have degree 2,
and it is possible to contract them all to get

K

1

,

3

;
such a

sequence

of

contractions

is in fact a

graph homeomorphism

.

□

Theorem 4

A finite tree with exactly four leaves
is homeomorphic to either

K

1

,

4

or two joint copies of

K

1

,

3

.

Proof.

If a vertex of degree 4 exists,
then no other vertex of degree greater than 2 can exist,
or the tree would have five or more leaves.
Otherwise, a vertex of degree 3 must exist,
or the tree would be a streamline with only two leaves;
by the handshake lemma, there must be another one,
and no more, or the tree would have at least five leaves.
All the other vertices, except the leaves, have degree 2,
and it is possible to contract them all
to get either

K

1

,

4

(if there is a vertex of degree 4)
or two joint copies of

K

1

,

3

(if there are two vertices of degree 3);
such a sequence of contractions is in fact a graph homeomorphism.

□

Title

handshake lemma

Canonical name

HandshakeLemma

Date of creation

2013-03-22 18:01:52

Last modified on

2013-03-22 18:01:52

Owner

Ziosilvio (18733)

Last modified by

Ziosilvio (18733)

Numerical id

10

Author

Ziosilvio (18733)

Entry type

Theorem

Classification

msc 05C99

Generated on Fri Feb 9 07:57:10 2018 by

LaTeXML
harmonic function

harmonic function

A real or complex-valued function

f

:

V

→

ℝ

or

f

:

V

→

ℂ

defined on the vertices

V

of a graph

G

=

(

V

,

E

)

is called

harmonic

at

v

∈

V

if its value at

v

is its

average value

at the neighbours of

v

:

f

⁢

(

v

)

=

1

deg

⁡

(

v

)

⁢

∑

{

u

,

v

}

∈

E

f

⁢

(

u

)

.

It is called harmonic

except on

A

, for some

A

⊆

V

, if it is harmonic at each

v

∈

V

∖

A

, and harmonic if it is harmonic at each

v

∈

V

.

Any harmonic

f

:

ℤ

n

→

ℝ

, where

ℤ

n

is the

n

-dimensional grid, is if below (or above). However, this is not necessarily true on other graphs.

Title

harmonic function

Canonical name

HarmonicFunction1

Date of creation

2013-03-22 15:09:27

Last modified on

2013-03-22 15:09:27

Owner

mathcam (2727)

Last modified by

mathcam (2727)

Numerical id

5

Author

mathcam (2727)

Entry type

Definition

Classification

msc 05C99

Generated on Fri Feb 9 07:57:12 2018 by

LaTeXML
homeomorphism

homeomorphism

We say that a graph

G

is

homeomorphic

to graph

H

if the
realization

R

⁢

(

G

)

of

G

is topologically homeomorphic (

http://planetmath.org/

Homeomorphism

) to

R

⁢

(

H

)

or, equivalently,

G

and

H

have

isomorphic

subdivisions

.

Adapted with permission of the author from

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

homeomorphism

Canonical name

Homeomorphism1

Date of creation

2013-03-22 12:31:55

Last modified on

2013-03-22 12:31:55

Owner

digitalis (76)

Last modified by

digitalis (76)

Numerical id

6

Author

digitalis (76)

Entry type

Definition

Classification

msc 05C99

Related topic

Subdivision

Related topic

Realization

Generated on Fri Feb 9 07:57:14 2018 by

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infinite graph

infinite graph

An is graph such that the edge and

vertex sets

each have

infinite

cardinality

. An example of an

infinite graph

is that of

ℤ

2

as stipulated in the entry

locally finite graph

.

Title

infinite graph

Canonical name

InfiniteGraph

Date of creation

2013-03-22 16:00:57

Last modified on

2013-03-22 16:00:57

Owner

sjm1979 (13837)

Last modified by

sjm1979 (13837)

Numerical id

5

Author

sjm1979 (13837)

Entry type

Definition

Classification

msc 05C99

Generated on Fri Feb 9 07:57:16 2018 by

LaTeXML
Kneser graphs

Kneser graphs

Let

k

,

n

be

positive integers

with

k

≤

n

. The

Kneser graph

K

n

:

k

has as its

vertex set

the

(

n

k

)

k

-element subsets of

{

1

,

2

,

…

,

n

}

. Two vertices are

adjacent

if and only if they correspond to disjoint subsets.

The graph

K

n

:

1

is the

complete graph

on

n

vertices. Another well-known Kneser graph is

K

5

:

2

, better known as the

Petersen graph

, which is shown in figure

1

. The Petersen graph is often a

counterexample

to graph-theoretical

conjectures

.

Figure 1:

The Petersen graph

Title

Kneser graphs

Canonical

name

KneserGraphs

Date of creation

2013-03-22 14:16:49

Last modified on

2013-03-22 14:16:49

Owner

justice (4961)

Last modified by

justice (4961)

Numerical id

5

Author

justice (4961)

Entry type

Definition

Classification

msc 05C99

Defines

Petersen graph

Generated on Fri Feb 9 07:57:18 2018 by

LaTeXML
locally finite graph

locally finite graph

\PMlinkescapephrase

locally finite

A

locally finite graph

is a graph in which every vertex has finite (

http://planetmath.org/Finite

) degree.
Note that any

finite graph

is locally finite;
however,

infinite graphs

can also be locally finite.
For example, consider the graph given by

ℤ

2

,
where the points are the vertices and the

line segments

of unit length that connect vertices are edges:

Generated on Fri Feb 9 07:57:24 2018 by

LaTeXML
loop

loop

In

graph theory

, a

loop

is an edge which joins a vertex
to itself, rather than to some other vertex. By

definition

,
a graph cannot contain a loop; a

pseudograph

, however, may contain
both

multiple edges

and

multiple

loops. Note that by some definitions,
a

multigraph

may contain multiple edges and no loops, while other texts
define a multigraph as a graph
allowing multiple edges and multiple loops.

In

algebra

, a

loop

is a

quasigroup

which contains an

identity element

.

Title

loop

Canonical

name

Loop

Date of creation

2013-03-22 12:14:08

Last modified on

2013-03-22 12:14:08

Owner

drini (3)

Last modified by

drini (3)

Numerical id

10

Author

drini (3)

Entry type

Definition

Classification

msc 05C99

Classification

msc 20N05

Related topic

Graph

Related topic

Pseudograph

Related topic

Quasigroup

Generated on Fri Feb 9 07:57:25 2018 by

LaTeXML
minor (of a graph)

minor (of a graph)

A graph

H

is a

minor

of

G

, written

G

≻

H

or

H

≺

G

, if it is a

subgraph

of a graph obtained from

G

by a sequence of edge-contractions.

Adapted with permission of the author from

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

minor (of a graph)

Canonical

name

MinorofAGraph

Date of creation

2013-03-22 12:31:46

Last modified on

2013-03-22 12:31:46

Owner

digitalis (76)

Last modified by

digitalis (76)

Numerical id

4

Author

digitalis (76)

Entry type

Definition

Classification

msc 05C99

Synonym

minor

Related topic

GraphMinorTheorem

Generated on Fri Feb 9 07:57:27 2018 by

LaTeXML
neighborhood (of a vertex)

neighborhood (of a vertex)

For a graph

G

, the set of vertices

adjacent

to a vertex

x

∈

G

, the

neighborhood

of

x

, is denoted by

Γ

⁢

(

x

)

. Occasionally one calls

Γ

⁢

(

x

)

the

open

neighborhood of

x

, and

Γ

∪

{

x

}

the

closed

neighborhood of

x

.

Adapted with permission of the author from

Modern

Graph Theory

by Béla Bollobás, published by Springer-Verlag New York, Inc., 1998.

Title

neighborhood (of a vertex)

Canonical

name

NeighborhoodofAVertex

Date of creation

2013-03-22 11:58:03

Last modified on

2013-03-22 11:58:03

Owner

digitalis (76)

Last modified by

digitalis (76)

Numerical id

10

Author

digitalis (76)

Entry type

Definition

Classification

msc 05C99

Synonym

neighborhood

Related topic

Graph

Generated on Fri Feb 9 07:57:29 2018 by

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