Geometric Geometric RepresentationsRepresentations of of Graphs Graphs

A survey of recent results and problems

Jan Kratochvíl, Prague

Outline of the Talk

Intersection Graphs Recognition of the Classes Sizes of Representations Optimization Problems Interval Filament Graphs Representations of Planar Graphs

Intersection Graphs

{Mu, u VG} uv EG Mu Mv

Interval graphs INT

Interval graphs INT

Circular Arc graphsCA

Interval graphs INT

Circular Arc graphsCA

Circle graphs CIR

Circular Arc graphsCA

Circle graphs CIR

Polygon-Circle graphs PC

SEG

SEG CONV

SEG CONV

STRING

INT

CA

CIR

PC

CONV

STR

SEG

2. Complexity of Recognition

Upper bound Lower bound

• P

• NP NP-hard

• PSPACE

• Decidable

• Unknown

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

Koebe 1990

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

Koebe 1990

J.K. 1991

J.K. 1991

J.K. 1991

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

Koebe 1990

J.K. 1991

J.K. 1991

J.K. 1991J.K., Matoušek 1994

K-M 1994

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

Koebe 1990

J.K. 1991

J.K. 1991

J.K. 1991J.K., Matoušek 1994

K-M 1994

Pach, Tóth 2001; Schaefer, Štefankovič 2001

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

Koebe 1990

J.K. 1991

J.K. 1991

J.K. 1991J.K., Matoušek 1994

K-M 1994

Schaefer, Sedgwick, Štefankovič 2002

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

Koebe 1990

J.K. 1991

J.K. 1991

J.K. 1991J.K., Matoušek 1994

K-M 1994

Schaefer, Sedgwick, Štefankovič 2002

?

?

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

J.K. 1991

J.K. 1991

J.K. 1991J.K., Matoušek 1994

K-M 1994

Schaefer, Sedgwick, Štefankovič 2002

?

?

?

Thm: Recognition of CONV graphs is in PSPACE

Reduction to solvability of polynomial inequalities in R:

x1, x2, x3 … xn R s.t.

P1(x1, x2, x3 … xn) > 0

P2(x1, x2, x3 … xn) > 0

…

Pm(x1, x2, x3 … xn) > 0 ?

{Mu, u VG} uv EG Mu Mv

Mu

Mv

Mw

Mz

Mu

Mv

Mw

Mz

Choose Xuv Mu Mv for every uv EG

Xuw

Xuz

Xuv

Cu Cv Mu Mv uv EG

Mu

Mv

Mw

Mz

Replace Mu by Cu = conv(Xuv : v s.t. uv EG) Mu

Xuw

Xuz

Xuv

Introduce variables xuv , yuv R s.t. Xuv = [xuv , yuv ] for uv EG

Introduce variables xuv , yuv R s.t. Xuv = [xuv , yuv ] for uv EG

uv EG Cu Cv guaranteed by the choice Cu = conv(Xuv : v s.t. uv EG)

Introduce variables xuv , yuv R s.t. Xuv = [xuv , yuv ] for uv EG

uv EG Cu Cv guaranteed by the choice Cu = conv(Xuv : v s.t. uv EG)

uv EG Cu Cv = separating lines

Introduce variables xuv , yuv R s.t. Xuv = [xuv , yuv ] for uv EG

uv EG Cu Cv guaranteed by the choice Cu = conv(Xuv : v s.t. uv EG)

uw EG Cu Cw = separating lines

Cu

Cw

auwx + buwy + cuw = 0

Introduce variables xuv , yuv R s.t. Xuv = [xuv , yuv ] for uv EG

uv EG Cu Cv guaranteed by the choice Cu = conv(Xuv : v s.t. uv EG)

uw EG Cu Cw = separating lines

Cu

Cw

auwx + buwy + cuw = 0

Representation is described by inequalities

(auwxuv + buwyuv + cuw) (auwxwz + buwywz + cuw) < 0 for all u,v,w,z s.t. uv, wz EG and uw EG

INT

CA

CIR

PC

CONV

STR

SEG

INT

CA

CIR

PC

CONV

STR

SEG

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

J.K. 1991

J.K. 1991

J.K. 1991J.K., Matoušek 1994

K-M 1994

Schaefer, Sedgwick, Štefankovič 2002

?

?

?

Polygon-circle graphs representable by polygons of bounded size

Polygon-circle graphs representable by polygons of bounded size

k-PC = Intersection graphs of convex k-gons inscribed to a circle

2-PC = CIR 3-PC 4-PC

Polygon-circle graphs representable by polygons of bounded size

k-PC = Intersection graphs of convex k-gons inscribed to a circle

2-PC = CIR 3-PC 4-PC

PC = k-PC

k=2

Example forcing large number of corners

Example forcing large number of corners

Example forcing large number of corners

3-PC

CIR = 2-PC

PC

4-PC

5-PC

3-PC

CIR = 2-PC

PC

4-PC

5-PC

J.K., M. Pergel 2003

?

Thm: For every k 3, recognition of k-PC graphs is NP-complete.

Proof for k = 3.Reduction from 3-edge colorability of cubic

graphs.For cubic G = (V,E), construct H = (W,F)

so that

’(G) = 3 iff H 3-PC

W = {u1, u2, u3, u4, u5, u6}

{ae, e E} {bv, v V}

F = {u1 u2, u2u3, u3u4, u4u5, u5u6 , u6u1}

{aebv, v e E}

{bubv, u,v V}

{bvui, v V, i = 2,4,6}

{u1, u2, u3, u4, u5, u6}

{u1, u2, u3, u4, u5, u6}

{ae, e E}

{u1, u2, u3, u4, u5, u6}

{ae, e E}

{u1, u2, u3, u4, u5, u6}

{ae, e E}

{bv, v V}

{u1, u2, u3, u4, u5, u6}

{ae, e E}

{bv, v V}

’(G) = 3 H 3-PC

{u1, u2, u3, u4, u5, u6}

{ae, e E}

{bv, v V}

’(G) > 3 H 3-PC

{u1, u2, u3, u4, u5, u6}

{ae, e E}

{bv, v V}

’(G) > 3 H 3-PC

3. Sizes of Representations

Membership in NP – Guess and verify a representation

Problem – The representation may be of exponential size

Indeed – for SEG and STRING graphs, NP-membership cannot be proven in this way

STRING graphs

STRING graphs

Abstract Topological Graphs

G = (V,E), R { ef : e,f E } is realizable if G has a drawing D in the plane such that for every two edges e,f E,

De Df ef R

G = (V,E), R = is realizable iff G is planar

Worst case functions

Str(n) = min k s.t. every STRING graph on n vertices has a representation with at most k crossing points of the curves

At(n) = min k s.t. every AT graph with n edges has a realization with at most k crossing points of the edges

Lemma: Str(n) and At(n) are polynomially equivalent

STRING graphs requiring large representations Thm (J.K., Matoušek 1991):

At(n) 2cn

Thm (Schaefer, Štefankovič 2001):

At(n) n2n-2

Sizes of SEG representations

Rational endpoints of segments Integral endpoints Size of representation = max coordinate of

endpoint (in absolute value)

Sizes of SEG representations

Thm (J.K., Matoušek 1994) For every n, there is a SEG graph Gn with O(n2) vertices such that every SEG representation has size at least

22n

Thm (Schaefer, Štefankovič 2001): At(n) n2n-2

Lemma: In every optimal representation of an AT graph, if an edge e is crossed by k other edges, then it carries at most 2k-1 crossing points.

e

e crossed by e1, e2, … , ek

e

e crossed by e1, e2, … , ek

(u1, u2, …, uk) - binary vector expressing the parity of the number of intersections of e and ei between the beginning of e and this location

e

e crossed by e1, e2, … , ek

(u1, u2, …, uk) - binary vector expressing the parity of the number of intersections of e and ei between the beginning of e and this location

If the number of crossing points on e is 2k, two of these vectors are the same

e

e crossed by e1, e2, … , ek

(u1, u2, …, uk) - binary vector expressing the parity of the number of intersections of e and ei between the beginning of e and this location

If the number of crossing points on e is 2k, two of this vectors are the same, and hence we find a segment on e where all other edges have even number of crossing points

e

e

e

e

2m crossing points with e4m crossing points with the circle

e

2m crossing points with e4m crossing points with the circle

e

2m crossing points with e4m crossing points with the circle

Circle inversion

e

2m crossing points with e4m crossing points with the circle

Circle inversion

Symmetric flip

e

2m crossing points with e4m crossing points with the circle

Circle inversion

Symmetric flip

2m crossing pointswith the circle, no newcrossing points arouse

e

2m crossing points with e4m crossing points with the circle

Circle inversion

Symmetric flip

2m crossing pointswith the circle, no newcrossing points arouse

Reroute e along the semicircle with fewernumber of crossing points

e

2m crossing points with e4m crossing points with the circle

Circle inversion

Symmetric flip

2m crossing pointswith the circle, no newcrossing points arouse

Reroute e along the semicircle with fewernumber of crossing points

Better realization - m < 2m

4. Optimization problems

INT

CA

CIR

PC

CONV

STR

SEG

Determining the chromatic number

INT

CA

CIR

PC

CONV

STR

SEG

(G) k for fixed k

INT

CA

CIR

PC

CONV

STR

SEG

Determining the independence number

INT

CA

CIR

PC

CONV

STR

SEG

Determining the clique number

J.K., Nešetřil1989

INT

CA

CIR

PC

CONV

STR

SEG

Determining the independence number - Interval filament graphs

IFAGavril 2000

Interval filament graphs

AA-mixed graphs

A A is a class of graphs.

G = (V,E) is AA-mixed if

E = E1 E2 and E2 is transitively oriented so that

xy E2 and yz E1 imply xz E1 , and

(V,E1) AA

Mixed condition

Thm (Gavril 2000): If WEIGHTED CLIQUE is polynomial in graphs from class A, A, then it is also polynomial in AA-mixed graphs.

Thm (Gavril 2000): If WEIGHTED CLIQUE is polynomial in graphs from class A, A, then it is also polynomial in AA-mixed graphs.

Thm (Gavril 2000):

CO-IFA = (CO-INT)-mixed

Thm (Gavril 2000): If WEIGHTED CLIQUE is polynomial in graphs from class A, A, then it is also polynomial in AA-mixed graphs.

Thm (Gavril 2000):

CO-IFA = (CO-INT)-mixedCorollary: WEIGHTED INDEPENDENT

SET is polynomial in IFA graphs

Interval filament graphs

INT

CA

CIR

PC

STR

INT

CA

CIR

PC

STR

Upper bound Lower bound

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

J.K. 1991Schaefer, Sedgwick, Štefankovič 2002

?

IFA IFA?

6. Representations of Planar Graphs

Problem (Pollack 1990): Planar SEG ? Known: Planar CONV Koebe: Planar graphs are exactly contact graphs of disks. Corollary: Planar 2-STRING Problem (Fellows 1988): Planar 1-STRING ? De Fraysseix, de Mendez (1997): Planar graphs are contact

graphs of triangles De Fraysseix, de Mendez (1997): 3-colorable 4-connected

triangulations are intersection graphs of segments Noy et al. (1999): Planar triangle-free graphs are in SEG

6. Representations of Co- Planar Graphs

J.K., Kuběna (1999): Co-Planar CONV Corollary: CLIQUE is NP-hard for CONV graphs Problem: Co-Planar SEG ?

Thank youThank you

6th International Czech-Slovak Symposium on

Combinatorics, Graph Theory, Algorithms and Applications

Prague, July 10-15, 2006

Honoring the 60th birthday of Jarik Nešetřil