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.
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