Geometric Graph Theory

stingymilitaryElectronics - Devices

Nov 27, 2013 (3 years and 9 months ago)

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Geometric

Graph

Theory



Geometric

G
raph

T
heory



R
egular

Hamiltonian

Bipartite

Planar

They

are
the

same

graph
!!!!!

Geometric

G
raph

T
heory

Geometric

graph

Vertices

=
points

in general position (no
three

of
them

are
collinear
)

Edges

=
straight
-
line
segments

Non
-
geometric

graph

Geometric

graph





Geometric

G
raph

T
heory



Two

e
dges

may

cross

each

other

Geometric

G
raph

T
heory

Crossing
-
free
geometric

graph

Theorem
.

(
Fáry
, 1948)

Every

planar

graph

admits

a
crossing
-
free
straight
-
line
drawing
.



Non
-
geometric

graph

Geometric

graph

Crossing
-
free
geometric

graph

Restrictions

on

the

drawing
:
fix

the

position of
the

points




Geometric

G
raph

T
heory

Theorem
.

(
Fáry
, 1948 )

Every

planar

graph

admits

a
crossing
-
free
straight
-
line
drawing
.



Restrictions

on

the

drawing
:
fix

the

position of
the

points




Question
:
Given

a
graph

G and a
point

set P, can
we

obtain

a
crossing
-
free
geometric

representation

of G
such

that

V(G)=P?

Not

always


for

planar

graphs

Geometric

G
raph

T
heory



Theorem
.

(
Gritzmann
,
Mohar
,
Pach
,

Pollack
, 1991)

Possible

for

outerplanar

graphs
.


Non
-
outerplanar

Question
:
Given

a
graph

G and a
point

set P, can
we

obtain

a
crossing
-
free
geometric

representation

of G
such

that

V(G)=P?

Outerplanar
: can
be

drawn

in
the

plane

without

crossings

so
that

all

the

vertices

lie
on

the

boundary

of
the

outer

face

Geometric

G
raph

T
heory



Unavoidable

crossings

Lemma

(
Pach
, 1999).
Any

geometric

graph

with

n>2
vertices

and e
edges

has at
least

e
-
3n+6
crossings
.


Question
:
Bounds

for

the

number

of
crossings

in G,
cr
(G).

n
.
n
e

.
9
0
75
33
1
cr(G)
2
3


….. and
much

more in:
János

Pach
.
Geometric

Graph

Theory
. In:
Surveys

in
Combinatorics

.
Lecture

Note Ser. 267, 1999, 167
-
200.

Geometric

G
raph

T
heory


Intersection

graphs

Vertices

=
objects

Edges

= non
-
empty

intersection

between

objects

Geometric

G
raph

T
heory



Vertices

=
objects

Edges

= non
-
empty

intersection

between

objects



Intersection

graph


Intersection

graphs

Geometric

G
raph

T
heory



Geometric

G
raph

T
heory



Geometric

G
raph

T
heory

Why

intersection

graphs
?



Every

graph

is

an

intersection

graph



Restricting

the

objects

by

geometrical

shape
,
there

are
different

motivations

and
applications
:


Geometric

G
raph

T
heory



Why

intersection

graphs
?



Every

graph

is

an

intersection

graph



Restricting

the

objects

by

geometrical

shape
,
there

are
different

motivations

and
applications
:


Wireless

networks
:
frequency

assignment

problems

Geometric

G
raph

T
heory



Map

labeling

Red points mark labels that could not be placed without intersections.

Geometric

G
raph

T
heory





Other

applications
:
scheduling
,
biology
, VLSI
desings
……..



Nice

characterizations
,
interesting

theoretical

properties
,
challenging

open
problems
……

Some

classes

of
intersection

graphs

Interval

graphs
,
segment

graphs
,
string

graphs
,
circle

graphs
, circular
arc

graphs
.

Geometric

G
raph

T
heory



Intersection

graph

of

a
collection

of
intervals


on

the

real line

Interval

graphs

Theorem

(
Booth
,
Lueker
, 1976).

Interval

graphs

can
be

recognized

in linear time.

Geometric

G
raph

T
heory



Segment

graphs

Theorem

(
Kratochvíl
,
Matousek
, 1994).

To

decide
whether

a
given

graph

can
be

represented

as a
segment

intersection

graph

is

NP
-
hard
.

Intersection

graph


of a set of
segments


Geometric

G
raph

T
heory

Theorem

(
Chalopin
,
Goncalves
, 2009).

Every

planar

graph

is

the

intersection

graph

of a set of line
segments

in
the

plane
.





Conjectured

by


Scheinerman

in 1984

Geometric

G
raph

T
heory



Theorem

(De Castro, Cobos, Dana, Márquez,
Noy
, 2002).

Every

triangle
-
free
planar

graph

is

the

intersection

graph

of a set
of line
segments

in
the

plane

in
three

directions
.

Geometric

G
raph

T
heory

String

graphs

Intersection

graph

of

simple curves in
the

plane

Theorem

(
Schaefer
,
Sedgwick
,
Stefankovic
, 2003).

To

decide
whether

a
given

graph

can
be

represented

as a
string

intersection

graph

is

NP
-
complete
.

Geometric

G
raph

T
heory

Theorem

(
Chalopin
,
Goncalves
,
Ochem
, 2007)

Every planar graph has a string representation in which
each pair of strings has at most one crossing point.

Geometric

G
raph

T
heory

Circle

graphs

Circular
arc

graphs

Geometric

G
raph

T
heory



Geometric

G
raph

T
heory

A
variation
:
Contact

graphs

The objects are not allowed to ”cross”, but only to “touch” each other.

Geometric

G
raph

T
heory



A
variation
:
Contact

graphs

The objects are not allowed to ”cross”, but only to “touch” each other.

Disk
Contact

Graph

or

Coin

Graph


Theorem

(
Koebe
, 1936).

G
is

a
coin

graph

if

and
only

if

it

is

planar
.

Corollary
. A
coin

graph

can
be

recognized


in linear time.