Geometric Representations of Graphs

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Geometric Representations of Graphs

• Jan Kratochvíl, DIMATIA, Prague

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Intersection Graphs
Mu, u ? VG uv ? EG ? Mu
? Mv ? ?
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String Graphs
Mu, u ? VG uv ? EG ? Mu
? Mv ? ?
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Personal Recollections

• 1982 Czech-Slovak Graph Theory, Prague

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Personal Recollections

• 1982 Czech-Slovak Graph Theory, Prague
• 1983 Prague
• 1990 Tempe, Arizona

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Personal Recollections

• 1982 Czech-Slovak Graph Theory, Prague
• 1983 Prague
• 1990 Tempe, Arizona
• 1988 Bielefeld, Germany

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Intersection Graphs

• Every graph is an intersection graph.

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Intersection Graphs

• Every graph is an intersection graph.

Mu e ? EG u ? e
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Intersection Graphs

• Every graph is an intersection graph.

Mu e ? EG u ? e
uv ? EG ? Mu ? Mv ? ?
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Intersection Graphs

• Every graph is an intersection graph
• Restricting the sets

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Intersection Graphs

• Every graph is an intersection graph
• Restricting the sets by geometrical shape
• Motivation and applications in scheduling,
  biology, VLSI designs

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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

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Few Examples
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Few Examples

• Interval graphs -
• Gilmore, Hoffman 1964
• Fulkerson, Gross 1965
• Booth, Lueker 1975
• Trotter, Harary 1979

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Few Examples

• Interval graphs -
• - neat characterization
• chordal co-comparability
• - recognizble in linear time
• - most optimization
• problems solvable in polynomial time
• - perfect

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Few Examples

• SEG graphs -
• Ehrlich, Even, Tarjan 1976
• Scheinerman
• Erdös, Gyarfás 1987
• JK, Neetril 1990
• JK, Matouek 1994
• Thomassen 2002

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Few Examples

• SEG graphs -
• - recognition NP-hard and
• in PSPACE,
• NP-membership open
• - coloring, independent
• set NP-hard, complexity of CLIQUE open
• - near-perfectness open

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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.

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Few Examples

• String graphs -
• Sinden 1966
• Ehrlich, Even, Tarjan 1976
• JK 1991
• JK, Matouek 1991
• Pach, Tóth 2001
• tefankovic, Schaffer 2001, 2002

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Few Examples

• CONV graphs -
• Ogden, Roberts 1970
• JK, Matouek 1994
• Agarwal, Mustafa 2004
• Kim, Kostochka,
• Nakprasit 2004

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Few Examples

• PC graphs -
• Fellows 1988
• Koebe 1990
• JK, Kostochka 1994
• Spinrad
• JK, Pergel 2002
• Pergel 2007

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Few Examples

• Circle graphs -
• De Fraysseix 1984
• Bouchet 1985
• Gyarfas 1987
• Unger 1988
• Kloks 1993
• Kostochka 1994

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Few Examples

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

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Few Examples

• Circular Arc graphs -
• Tucker 1971, 1980
• Gavril 1974
• Gyarfás 1987
• Spinrad 1988
• Hell, Bang-Jensen, Huang 1990

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Few Examples

• Circular Arc graphs -
• Tucker 1971, 1980
• Gavril 1974
• Gyarfás 1987
• Spinrad 1988
• Hell, Bang-Jensen, Huang 1990

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Outline

• String graphs
• CONV and PC graphs
• Representations of planar graphs

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1. String graphs

• Sinden 1966

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1. String graphs

• Sinden 1966 IG(regions)

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1. String graphs

• Sinden 1966 IG(regions)
• Graham 1974

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1. String graphs

• Sinden 1966
• JK, Goljan, Kucera 1982

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1. String graphs

• Sinden 1966
• JK, Goljan, Kucera 1982
• Thomas 1988
• IG(topologically con)
• all graphs,
• String IG(arc-connected sets)

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1. String graphs

• Sinden 1966
• JK, Goljan, Kucera 1982
• Thomas 1988
• JK 1991 NP-hard

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1. String graphs
STRING
SEG
CONV
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1. String graphs
STRING
SEG
CONV
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1. String graphs

• Sinden 1966
• JK, Goljan, Kucera 1982
• Thomas 1988
• JK 1991 NP-hard
• Recognition in NP?

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1. String graphs

• Sinden 1966
• JK, Goljan, Kucera 1982
• Thomas 1988
• JK 1991 NP-hard
• Recognition in NP?

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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

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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

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String graphs requiring large representations

• Thm (J.K., Matouek 1991)
• At(n) ? 2cn

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1. String graphs

• Sinden 1966
• JK, Goljan, Kucera 1982
• Thomas 1988
• JK 1991 NP-hard
• Recognition in NP?
• Are they recognizable at all?

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Thm (Pach, Tóth 2001) At(n)
? nn Thm (Schaefer, tefankovic 2001)
At(n) ? n2n-2
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1. String graphs

• Sinden 1966
• JK, Goljan, Kucera 1982
• JK 1991 NP-hard
• Schaefer, Sedgwick,
• tefankovic 2002
• String graph recognition is in NP (Lempel-Ziv
  compression)

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1. Some subclasses
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1. Some subclasses
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1. Some subclasses

• Complements of
• Comparability graphs
• (Golumbic 1977)

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Co-comparability graphs
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Co-comparability graphs
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Co-comparability graphs
? ?
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Co-comparability graphs
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Co-comparability graphs
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1. Some subclasses

• Zwischenring graphs
• NP-hard
• (Middendorf, Pfeiffer)

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1. Some subclasses

• Outerstring graphs
• NP-hard
• (Middendorf, Pfeiffer)

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1. Some subclasses

• Outerstring graphs
• NP-hard
• (Middendorf, Pfeiffer)

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1. Some subclasses

• Interval filament graphs
• (Gavril 2000)
• CLIQUE and IND SET
• can be solved in
• polynomial time

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2. CONV and PC

• JK, Matouek 1994
• recognition in PSPACE

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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) gt 0
• P2(x1, x2, x3 xn) gt 0
• Pm(x1, x2, x3 xn) gt 0 ?

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Mv
Mw
Mu
Mz
Mu, u ? VG uv ? EG ? Mu
? Mv ? ?
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Mv
Xuv
Xuw
Mw
Mu
Xuz
Mz
Choose Xuv ? Mu ? Mv for every uv ? EG
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Mv
Xuv
Xuw
Mw
Mu
Xuz
Mz
Replace Mu by Cu conv(Xuv v s.t. uv ? EG) ?
Mu
Cu ? Cv ? ? ? Mu ? Mv ? ? ? uv
? EG
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Introduce variables xuv , yuv ? R s.t. Xuv
xuv , yuv for uv ? EG
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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)
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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
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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
Cw
Cu
auwx buwy cuw 0
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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
Cw
Xwz
Cu
Xuv
auwx buwy cuw 0
Representation is described by inequalities
(auwxuv buwyuv cuw) (auwxwz buwywz cuw) lt
0 for all u,v,w,z s.t.
uv, wz ? EG and uw ? EG
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2. Recognition NP-membership

• Guess and verify

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2. Recognition NP-membership

• Guess and verify
• INT, CA, CIR, PC, Co-Comparability
• IFA mixing characterization
• - CONV, SEG ?
• !! String Lempel-Ziv compression

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2. Recognition NP-membership

• Thm (JK, Matouek 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).

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2. CLIQUE in CONV graphs

• CO-PLANAR ? CONV (JK, Kubena 99)

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2. CLIQUE in CONV graphs

• CO-PLANAR ? CONV (JK, Kubena 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, Neetril
  1990)

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2. CLIQUE in MAX-TOL graphs
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2. MAX-TOLERANCE
(Golumbic, Trenk 2004)
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2. MAX-TOLERANCE
S Iu u ? VG intervals, tu ? R
tolerances uv ? EG iff Iu ? Iv max tu,
tv

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

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2. MAX-TOLERANCE
Tu
Tv
tu
Iu
Iv

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

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A
B
C
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Maximal cliques
Q a maximal clique
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Maximal cliques
Q a maximal clique
t
h
v

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

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Maximal cliques
Q a maximal clique
t
h
v

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

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• Claim Q(h,v,t) Q

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• Claim Q(h,v,t) Q
• Proof
• 1) Q ? Q(h,v,t)

h
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• Claim Q(h,v,t) Q
• Proof
• 1) Q ? Q(h,v,t)
• 2) Q(h,v,t) is a clique

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• 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.

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• Claim Q(h,v,t) Q
• Proof
• 1) Q ? Q(h,v,t)
• 2) Q(h,v,t) is a clique

a
b
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• 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
h
b
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Maximal cliques
Q a maximal clique
t
h
v

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

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2. Polygon-circle graphs

• PC graphs -
• Fellows 1988
• Koebe 1990
• JK, Kostochka 1994
• Spinrad
• JK, Pergel 2002
• Pergel 2007

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2. Polygon-circle graphs

• PC graphs -
• Fellows 1988
• Koebe 1990
• JK, Kostochka 1994
• Spinrad
• JK, Pergel 2002
• Pergel 2007

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2. Polygon-circle graphs

• PC graphs -
• Fellows 1988
• Koebe 1990
• JK, Kostochka 1994
• Spinrad
• JK, Pergel 2002
• Pergel 2007

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2. Polygon-circle graphs
IFA
CA
CIR
PC
CHOR
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2. Polygon-circle graphs
IFA
CA
CIR
PC
CHOR
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2. Polygon-circle graphs
IFA
CA
CIR
PC
CHOR
Pergel 2007
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2. Polygon-circle graphs
IFA
CA
CIR
PC
CHOR
Pergel 2007
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2. Short cycles

• Do short cycles help?

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2. Short cycles

• Do short cycles mind?
• Does large
  girth help?

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UNIT-DISK
DISK
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UNIT-DISK
DISK
PSEUDO-DISK
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2. Short cycles

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

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2. Short cycles

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

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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.

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Koebe (1936)
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2. Short cycles

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

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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.

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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.

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3. Representations of planar graphs
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3. Representations of planar graphs
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3. Representations of planar graphs

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

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3. Representations of planar graphs

• Planar graphs are exactly contact graphs of disks
  (Koebe 1934)
• PLANAR ? DISK
• PLANAR ? CONV
• PLANAR ? 2-STRING

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3. Representations of planar graphs

• PLANAR ? 2-STRING
• Problem (Fellows 1988) Planar ? 1-STRING ?
• True Chalopin, Gonçalves, and Ochem
• SODA 2007

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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, )

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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.)

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3. Bipartite planar graphs
De Fraysseix, Ossona de Mendez, Pach
f
f
e
3
5
e
d
d
c
6
7
2
c
1
4
b
b
a
a
1
2
3
4
5
6
7
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3. Bipartite planar graphs
De Fraysseix, Ossona de Mendez, Pach
f
e
d
c
b
a
1
2
3
4
5
6
7
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3. Bipartite planar graphs
De Fraysseix, Ossona de Mendez, Pach
f
e
d
c
b
a
1
2
3
4
5
6
7
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3. Representations of planar graphs

• PLANAR ? CONV
• Planar graphs are contact graphs of triangles (de
  Fraysseix, Ossona de Mendez 1997)

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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?

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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

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3. Representations of planar graphs
2
1
b
a
c
3
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3. Representations of planar graphs
2
1

• 1

b
a
a
b
c
c
3
2
3
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3. Representations of planar graphs
2
1

• 1

b
a
a
b
c
c
3
2
3
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3. Planar open problems

• PLANAR ? MAX-TOL? (Lehmann)
• (i.e. are planar graphs intersection graphs
  of homothetic triangles?)

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3. Planar open problems

• PLANAR ? MAX-TOL? (Lehmann)
• Conjecture (Felsner, JK 2007) Planar
• 4-connected triangulations are contact
  graphs of homothetic triangles.

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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.

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3. Representations of planar graphs
2
1
b
a
c
3
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3. Representations of planar graphs
2
1
b
a
a
b
c
c
3
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3. Representations of planar graphs
2
1
b
a
a
b
c
c
3
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4. Invitation

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

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HAPPY BIRTHDAY Tom !