Geometric Representations of Graphs Jan Kratochvíl, DIMATIA, Prague.

Geometric Representations Geometric Representations of of GraphsGraphs

Jan Kratochvíl, DIMATIA, Prague

Intersection Graphs

{Mu, u VG} uv EG Mu Mv

String Graphs


Personal Recollections

1982 – Czech-Slovak Graph Theory, Prague



1983 – Prague

1990 – Tempe, Arizona



1983 – Prague

1990 – Tempe, Arizona

1988 – Bielefeld, Germany

Intersection Graphs

Every graph is an intersection graph.

Intersection Graphs


Mu = {e EG | u e}

Intersection Graphs


uv EG Mu Mv

Mu = {e EG | u e}

Intersection Graphs

Every graph is an intersection graph

Restricting the sets

Intersection Graphs


Restricting the sets – by geometrical shape

Motivation and applications in scheduling, biology, VLSI designs …

Intersection Graphs


Restricting the sets – by geometrical shape

Motivation and applications in scheduling, biology, VLSI designs …

Nice characterizations, interesting theoretical properties, challenging open problems

Few Examples

Few Examples

Interval graphsInterval graphs -

Gilmore, Hoffman 1964

Fulkerson, Gross 1965

Booth, Lueker 1975

Trotter, Harary 1979

…

Few Examples

Interval graphsInterval graphs -

- neat characterization

chordal + co-comparability

- recognizble in linear time

- most optimization

problems solvable in polynomial time

- perfect

Few Examples

SEG graphsSEG graphs -

Ehrlich, Even, Tarjan 1976

Scheinerman

Erdös, Gyarfás 1987

JK, Nešetřil 1990

JK, Matoušek 1994

Thomassen 2002

Few Examples

SEG graphsSEG graphs -

- recognition NP-hard and

in PSPACE,

NP-membership open

- coloring, independent

set NP-hard, complexity of CLIQUE open

- near-perfectness open

Near-perfect graph classes

A graph class G is near-perfect if there exists a function f such that

(G) f((G))

for every G G.

Few Examples

String graphsString graphs -

Sinden 1966

Ehrlich, Even, Tarjan 1976

JK 1991

JK, Matoušek 1991

Pach, Tóth 2001

Štefankovič, Schaffer 2001, 2002

Few Examples

CONV graphsCONV graphs -

Ogden, Roberts 1970

JK, Matoušek 1994

Agarwal, Mustafa 2004

Kim, Kostochka,

Nakprasit 2004

Few Examples

PC graphsPC graphs -

Fellows 1988

Koebe 1990

JK, Kostochka 1994

Spinrad

JK, Pergel 2002

Pergel 2007

Few Examples

Circle graphsCircle graphs -

De Fraysseix 1984

Bouchet 1985

Gyarfas 1987

Unger 1988

Kloks 1993

Kostochka 1994

Few Examples

Circle graphsCircle graphs -- recognizable in linear time - coloring NP-hard- independent set, cliquesolvable in polynomial time- near-perfect log O(2) - close bounds open

Few Examples

Circular Arc graphsCircular Arc graphs -

Tucker 1971, 1980

Gavril 1974

Gyarfás 1987

Spinrad 1988

Hell, Bang-Jensen, Huang 1990

…

Outline

String graphs CONV and PC graphs Representations of planar graphs

1. String graphs

Sinden 1966

1. String graphs

Sinden 1966 = IG(regions)

1. String graphs

Sinden 1966 = IG(regions)

Graham 1974

1. String graphs

Sinden 1966

JK, Goljan, Kučera 1982

1. String graphs

Sinden 1966


Thomas 1988

IG(topologically con) =

all graphs,

String = IG(arc-connected sets)

1. String graphs

Sinden 1966


Thomas 1988

JK 1991 – NP-hard

1. String graphs

SEGCONV

STRING

1. String graphs

Sinden 1966

JK, Goljan, Kucera 1982

Thomas 1988

JK 1991 – NP-hard

Recognition in NP?

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

1. String graphs

Sinden 1966

JK, Goljan, Kucera 1982

Thomas 1988

JK 1991 – NP-hard

Recognition in NP?

Are they recognizable at all?

Thm (Pach, Tóth 2001): At(n) nn

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

1. String graphs

Sinden 1966


JK 1991 – NP-hard

Schaefer, Sedgwick,

Štefankovič 2002 –

String graph recognition is in NP (Lempel-Ziv compression)

1. Some subclasses

1. Some subclasses

Complements of

Comparability graphs

(Golumbic 1977)

Co-comparability graphs


=

1. Some subclasses

“Zwischenring” graphs

NP-hard

(Middendorf, Pfeiffer)

1. Some subclasses

Outerstring graphs

NP-hard

(Middendorf, Pfeiffer)

1. Some subclasses

Interval filament graphs

(Gavril 2000)

CLIQUE and IND SET

can be solved in

polynomial time

2. CONV and PC

JK, Matoušek 1994 –

recognition in PSPACE

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

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


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




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

Xuv

Xwz

2. Recognition – NP-membership

“Guess and verify”


“Guess and verify”

- INT, CA, CIR, PC, Co-Comparability

- IFA – mixing characterization

- CONV, SEG ?

!! String – Lempel-Ziv compression


Thm (JK, Matoušek 1994): For every n there is a graph Gn SEG with O(n2) vertices s.t. every SEG representation with integer endpoints has a coordinate of absolute value 22n.

Same for CONV (Pergel 2008).

2. CLIQUE in CONV graphs

- CO-PLANAR CONV (JK, Kuběna 99)

2. CLIQUE in CONV graphs

- CO-PLANAR CONV (JK, Kuběna 99)

- Corollary: CLIQUE is NP-complete for CONV graphs. (Since INDEPENDENT SET is NP-complete for planar graphs.)

- CLIQUE in SEG graphs still open (JK, Nešetřil 1990)

2. CLIQUE in MAX-TOL graphs

2. MAX-TOLERANCE

(Golumbic, Trenk 2004)

2. MAX-TOLERANCE

S S = {Iu | u VG } intervals, tu RR tolerances

uv EG iff |Iu Iv| ≥ max {tu, tv}

2. MAX-TOLERANCE

Theorem (Kaufmann, JK, Lehmann, Subramarian, 2006): Max-tolerance graphs are exactly intersection graphs of homothetic triangles (semisquares)

2. MAX-TOLERANCE

Iu

tu

Tu

Iv

Tv

Lemma (folklore): Disjoint convex polygons are separated by a line parallel to a side of one of them.

A B

C

Maximal cliques

Q a maximal clique

Maximal cliques

h highest basis of Q, v rightmost vertical side,t lowest diagonal side

Q a maximal cliquet

h v

Maximal cliques

Q(h,v,t) = all triangles that intersect h,v and t

Q a maximal cliquet

h v

Claim: Q(h,v,t) = Q

Claim: Q(h,v,t) = Q

Proof:

1) Q Q(h,v,t)

h

Claim: Q(h,v,t) = Q

Proof:

1) Q Q(h,v,t)

2) Q(h,v,t) is a clique

Claim: Q(h,v,t) = Q

Proof:

1) Q Q(h,v,t)


Suppose a,b Q(h,v,t) are disjoint, hence separated by a line parallel to one of the sides, say horizontal.

Claim: Q(h,v,t) = Q

Proof:

1) Q Q(h,v,t)


a

b

Claim: Q(h,v,t) = Q

Proof:

1) Q Q(h,v,t)


b cannot intersect h,

a contradiction

a

b

h

Maximal cliques

Q(h,v,t) = all triangles that intersect h,v and tHence G has O(n3) maximal cliques.

Q a maximal cliquet

h v

2. Polygon-circle graphs

PC graphsPC graphs -

Fellows 1988

Koebe 1990

JK, Kostochka 1994

Spinrad

JK, Pergel 2002

Pergel 2007


CIR PC

IFA

CA

CHOR


CIR PC

IFA

CA

CHOR

Pergel 2007

2. Short cycles

Do short cycles help?

2. Short cycles

Do short cycles mind?

Does large girth help?

DISKUNIT-DISK

DISKUNIT-DISK

PSEUDO-DISK

2. Short cycles

Thm (J.K. 1996) Triangle-free intersection graphs of pseudodisks are planar.

2. Short cycles

Thm (J.K. 1996) Triangle-free intersection graphs of pseudodisks are planar.

Corollary: Recognition of triangle-free PSEUDO-DISK and DISK graphs is polynomial.

Koebe (1936)

2. Short cycles

Thm (J.K. 1996) Triangle-free STRING graphs are NP-hard to recognize.

2. Short cycles

Thm (J.K. 1996) Triangle-free STRING graphs are NP-hard to recognize.

Thm (JK, Pergel 2007) PC graphs of girth 5 can be recognized in polynomial time.

Thm (JK, Pergel 2007) For each k, recognition of SEG graphs of girth k is NP-hard.

2. Short cycles

Problem: Is recognition of String graphs of girth k NP-complete for every k ?

Thm (JK, Pergel 2007) PC graphs of girth 5 can be recognized in polynomial time.

Thm (JK, Pergel 2007) For each k, recognition of SEG graphs of girth k is NP-hard.

3. Representations of planar graphs


- Planar graphs are exactly contact graphs of disks (Koebe 1934)


- Planar graphs are exactly contact graphs of disks (Koebe 1934)

- PLANAR DISK

- PLANAR CONV

- PLANAR 2-STRING


- PLANAR 2-STRING- Problem (Fellows 1988): Planar 1-STRING ?- True: Chalopin, Gonçalves, and Ochem

[SODA 2007]


- PLANAR 2-STRING- Problem (Fellows 1988): Planar 1-STRING ?- True: Chalopin, Gonçalves, and Ochem

[SODA 2007]

- Problem: PLANAR SEG? (Pollack, Scheinerman, West, …)


- PLANAR SEG (?)- 3-colorable 4-connected triangulations are

intersection graphs of segments (de Fraysseix, de Mendez 1997)

- Planar triangle-free graphs are in SEG (Noy et al. 1999)

- Planar bipartite graphs are grid intersection (Hartman et al. 91; Albertson; de Fraysseix et al.)

3. Bipartite planar graphsDe Fraysseix, Ossona de Mendez, Pach

d

c

f

e

b

a

12

3 5

6

4

7

abcdef

1 2 3 4 5 6 7

3. Bipartite planar graphsDe Fraysseix, Ossona de Mendez, Pach

abcdef

1 2 3 4 5 6 7


- PLANAR CONV- Planar graphs are contact graphs of triangles (de

Fraysseix, Ossona de Mendez 1997)



Fraysseix, Ossona de Mendez 1997)- Are planar graphs contact graphs of homothetic

triangles?



Fraysseix, Ossona de Mendez 1997)- Are planar graphs contact graphs of homothetic

triangles?- No


1 2

3

b

c

a


1 2

3

b

c

a

1

23

a b

c

3. Planar – open problems

- PLANAR MAX-TOL? (Lehmann)

(i.e. are planar graphs intersection graphs of homothetic triangles?)



- Conjecture (Felsner, JK 2007): Planar

4-connected triangulations are contact graphs of homothetic triangles.



- Conjecture (Felsner, JK 2007): Planar 4-connected triangulations are contact

graphs of homothetic triangles. This would imply that planar graphs are

intersection graphs of homothetic triangles.


1 2

3

b

c

aa b

c

4. Invitation

Graph Drawing, Crete, Sept 21 – 24, 2008 Prague MCW, July 28 – Aug 1, 2008

Geometric Representations of Graphs Jan Kratochvíl, DIMATIA, Prague.

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