Geometric Representations Geometric Representations of of GraphsGraphs

Jan Kratochvíl, DIMATIA, Prague

Intersection Graphs

{Mu, u VG} uv EG Mu Mv

String Graphs

{Mu, u VG} uv EG Mu Mv

Personal Recollections

1982 – Czech-Slovak Graph Theory, Prague

Personal Recollections

1982 – Czech-Slovak Graph Theory, Prague

1983 – Prague

1990 – Tempe, Arizona

Personal Recollections

1982 – Czech-Slovak Graph Theory, Prague

1983 – Prague

1990 – Tempe, Arizona

1988 – Bielefeld, Germany

Intersection Graphs

Every graph is an intersection graph.

Intersection Graphs

Every graph is an intersection graph.

Mu = {e EG | u e}

Intersection Graphs

Every graph is an intersection graph.

uv EG Mu Mv

Mu = {e EG | u e}

Intersection Graphs

Every graph is an intersection graph

Restricting the sets

Intersection Graphs

Every graph is an intersection graph

Restricting the sets – by geometrical shape

Motivation and applications in scheduling, biology, VLSI designs …

Intersection Graphs

Every graph is an intersection graph

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

…

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

JK, Goljan, Kučera 1982

Thomas 1988

IG(topologically con) =

all graphs,

String = IG(arc-connected sets)

1. String graphs

Sinden 1966

JK, Goljan, Kučera 1982

Thomas 1988

JK 1991 – NP-hard

1. String graphs

SEGCONV

STRING

1. String graphs

SEGCONV

STRING

1. String graphs

Sinden 1966

JK, Goljan, Kucera 1982

Thomas 1988

JK 1991 – NP-hard

Recognition in NP?

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, Goljan, Kučera 1982

JK 1991 – NP-hard

Schaefer, Sedgwick,

Štefankovič 2002 –

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

1. Some subclasses

1. Some subclasses

1. Some subclasses

Complements of

Comparability graphs

(Golumbic 1977)

Co-comparability graphs

Co-comparability graphs

Co-comparability graphs

=

Co-comparability graphs

Co-comparability graphs

1. Some subclasses

“Zwischenring” graphs

NP-hard

(Middendorf, Pfeiffer)

1. Some subclasses

Outerstring 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, 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)

uw EG Cu Cw = 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

Xuv

Xwz

2. Recognition – NP-membership

“Guess and verify”

2. Recognition – NP-membership

“Guess and verify”

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

- IFA – mixing characterization

- CONV, SEG ?

!! String – Lempel-Ziv compression

2. Recognition – NP-membership

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)

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

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)

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

a

b

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

Proof:

1) Q Q(h,v,t)

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

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

2. Polygon-circle graphs

PC graphsPC graphs -

Fellows 1988

Koebe 1990

JK, Kostochka 1994

Spinrad

JK, Pergel 2002

Pergel 2007

2. Polygon-circle graphs

PC graphsPC graphs -

Fellows 1988

Koebe 1990

JK, Kostochka 1994

Spinrad

JK, Pergel 2002

Pergel 2007

2. Polygon-circle graphs

CIR PC

IFA

CA

CHOR

2. Polygon-circle graphs

CIR PC

IFA

CA

CHOR

2. Polygon-circle graphs

CIR PC

IFA

CA

CHOR

Pergel 2007

2. Polygon-circle graphs

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.

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

3. Representations of planar graphs

3. Representations of planar graphs

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

3. Representations of planar graphs

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

- PLANAR DISK

- PLANAR CONV

- PLANAR 2-STRING

3. Representations of planar graphs

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

[SODA 2007]

3. Representations of planar graphs

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

[SODA 2007]

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

3. Representations of planar graphs

- 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

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

abcdef

1 2 3 4 5 6 7

3. Representations of planar graphs

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

Fraysseix, Ossona de Mendez 1997)

3. Representations of planar graphs

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

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

triangles?

3. Representations of planar graphs

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

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

triangles?- No

3. Representations of planar graphs

1 2

3

b

c

a

3. Representations of planar graphs

1 2

3

b

c

a

1

23

a b

c

3. Representations of planar graphs

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?)

3. Planar – open problems

- PLANAR MAX-TOL? (Lehmann)

- Conjecture (Felsner, JK 2007): Planar

4-connected triangulations are contact graphs of homothetic triangles.

3. Planar – open problems

- PLANAR MAX-TOL? (Lehmann)

- 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.

3. Representations of planar graphs

1 2

3

b

c

aa b

c

3. Representations of planar graphs

1 2

3

b

c

aa b

c

3. Representations of planar graphs

1 2

3

b

c

aa b

c

4. Invitation

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