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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
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