geometric representations of graphs

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

Circular Arc graphsCA

Interval graphs INT

Circle graphs CIR

Polygon-Circle graphs PC

SEG CONV

STRING

2. Complexity of Recognition

Upper bound Lower bound

• NP NP-hard

• PSPACE

• Decidable

• Unknown

Gilmore, Hoffman 1964

Tucker 1970

Bouchet 1985

Tucker 1970

Bouchet 1985

Koebe 1990

Tucker 1970

Bouchet 1985

Koebe 1990

J.K. 1991

Tucker 1970

Bouchet 1985

Koebe 1990

J.K. 1991

J.K. 1991J.K., Matoušek 1994

K-M 1994

Tucker 1970

Bouchet 1985

Koebe 1990

J.K. 1991

K-M 1994

Pach, Tóth 2001; Schaefer, Štefankovič 2001

Tucker 1970

Bouchet 1985

Koebe 1990

J.K. 1991

K-M 1994

Schaefer, Sedgwick, Štefankovič 2002

Tucker 1970

Bouchet 1985

Koebe 1990

J.K. 1991

K-M 1994

Tucker 1970

Bouchet 1985

J.K. 1991

K-M 1994

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

Choose Xuv Mu Mv for every uv EG

Cu Cv Mu Mv uv EG

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

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

uw EG Cu Cw = separating lines

auwx + buwy + cuw = 0

uw EG Cu Cw = separating lines

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

Tucker 1970

Bouchet 1985

J.K. 1991

K-M 1994

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

Example forcing large number of corners

CIR = 2-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}

{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

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

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 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

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

2m crossing points with e4m crossing points with the circle

Circle inversion

Symmetric flip

Circle inversion

Symmetric flip

2m crossing pointswith the circle, no newcrossing points arouse

Circle inversion

Symmetric flip

Reroute e along the semicircle with fewernumber of crossing points

Circle inversion

Symmetric flip

Reroute e along the semicircle with fewernumber of crossing points

Better realization - m < 2m

4. Optimization problems

Determining the chromatic number

(G) k for fixed k

Determining the independence number

Determining the clique number

J.K., Nešetřil1989

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):

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

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

geometric representations of graphs

uv eguv

uv egintroduce variables

choice cu

recognition of conv

x3 xn r

separating linescucwauwx

problemsjan kratochvl

survey of recent results

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