planar graph

planar graph

\xyoption

all

Description

A

planar graph

is a graph which can be drawn on a plane (a flat 2-d

surface

) or on a sphere, with no edges crossing. When drawn on a sphere, the edges divide its area in a number of regions called faces (or “countries”, in the context of

map coloring

). When drawn on a plane, there is one

outer country

taking up all the space outside the drawing. Every graph drawn on a sphere can be drawn on a plane (puncture the sphere in the

interior

of any one of the countries) and vice versa. Statements on map coloring are often simpler in terms of a spherical map because the outer country is no longer a special case.

The number of faces (countries) equals

c

+

1

where

c

is the

cyclomatic number

,

c

=

m

-

n

+

k

(where

m

is the number of edges,

n

the number of vertices, and

k

the number of

connected components

of the graph). All this holds equally for planar

multigraphs

and

pseudographs

.

No

complete graphs

above

K

4

are planar.

K

4

, drawn without crossings, looks like :

\xymatrix

⁢

&

⁢

A

⁢

\ar

⁢

@

-

[

d

]

⁢

\ar

⁢

@

-

[

d

⁢

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

d

⁢

r

]

⁢

&

⁢

&

⁢

B

⁢

\ar

⁢

@

-

[

d

⁢

l

]

⁢

\ar

⁢

@

-

[

d

⁢

r

]

⁢

&

⁢

C

⁢

\ar

⁢

@

-

[

r

⁢

r

]

⁢

&

⁢

&

⁢

D

Hence it is planar (try this for

K

5

.)

A drawing of a planar graph is a drawing in which each edge is drawn as a

line segment

. Every planar graph has a drawing. This result was found independently by Wagner, Fáry and Stein. Schnyder improved this further by showing how to draw any planar graph with

n

vertices on an integer grid of

O

⁢

(

n

2

)

area.

Definition

Ideally, this

definition

would just formalize what was described above. It will not, exactly. It will formalize the notion of a graph

with an

embedding

into the plane

.

Definition

.

Let

M

be a

topological manifold

. Then a

graph on

M

is a pair

(

G

,

ι

)

, where

1.

G

is a multigraph,

2.

ι

is a function from the

graph topology

of

G

into

M

, and

3.

ι

is a

homeomorphism

onto its image.

A

plane graph

is a graph on

ℝ

2

.

A

planar graph

is a graph

G

that has an embedding

ι

making

(

G

,

ι

)

into a plane graph.

Normally, the only

manifolds

M

that are of

interest

are two-dimensional.

The most usual question for which this definition finds a use is “can the following graph

G

be made into a graph on

M

?”. When

M

is the plane, this is usually phrased as “Is

G

a planar graph?”. Wagner’s

theorem

provides a criterion for answering this question. When

M

is a torus, the answer changes: the

complete bipartite graph

K

3

,

3

can be made into a graph on the torus.

A graph on a manifold has a notion of “face” as well as the usual graph notions of vertex and edge.

Title

planar graph

Canonical

name

PlanarGraph

Date of creation

2013-03-22 12:17:37

Last modified on

2013-03-22 12:17:37

Owner

archibal (4430)

Last modified by

archibal (4430)

Numerical id

12

Author

archibal (4430)

Entry type

Definition

Classification

msc 05C10

Synonym

planar

Related topic

WagnersTheorem

Related topic

KuratowskisTheorem

Related topic

CrossingLemma

Related topic

CrossingNumber

Related topic

FourColorConjecture

Defines

plane graph

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