Geometric Representations of Graphs

1
Geometric Representations of Graphs

• A survey of recent results and problems
• Jan Kratochvíl, Prague

2
Outline of the Talk

• Intersection Graphs
• Recognition of the Classes
• Sizes of Representations
• Optimization Problems
• Interval Filament Graphs
• Representations of Planar Graphs

3
Intersection Graphs
Mu, u ? VG uv ? EG ? Mu
? Mv ? ?
4
Interval graphs INT
5
Interval graphs INT
Circular Arc graphs CA
6
Interval graphs INT
Circular Arc graphs CA
Circle graphs CIR
7
Polygon-Circle graphs PC
Circular Arc graphs CA
Circle graphs CIR
8
SEG
9
CONV
SEG
10
CONV
SEG
STRING
11
STR
CONV
SEG
PC
CIR
CA
INT
12
2. Complexity of Recognition

• Upper bound Lower bound
• P
• NP NP-hard
• PSPACE
• Decidable
• Unknown

13
Lower bound
Upper bound
STR
STR
CONV
CONV
SEG
SEG
PC
PC
CIR
CIR
CA
CA
INT
INT
14
Lower bound
Upper bound
STR
STR
CONV
CONV
SEG
SEG
PC
PC
CIR
CIR
CA
CA
INT
INT
15
Lower bound
Upper bound
STR
STR
CONV
CONV
SEG
SEG
PC
PC
CIR
CIR
CA
CA
INT
INT
Gilmore, Hoffman 1964
16
Lower bound
Upper bound
STR
STR
CONV
CONV
SEG
SEG
PC
PC
CIR
CIR
CA
CA
INT
INT
Gilmore, Hoffman 1964
17
Lower bound
Upper bound
STR
STR
CONV
CONV
SEG
SEG
PC
PC
CIR
CIR
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
18
Lower bound
Upper bound
STR
STR
CONV
CONV
SEG
SEG
PC
PC
CIR
CIR
Bouchet 1985
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
19
Lower bound
Upper bound
STR
STR
CONV
CONV
SEG
SEG
PC
PC
Koebe 1990
CIR
CIR
Bouchet 1985
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
20
Lower bound
Upper bound
STR
STR
J.K. 1991
CONV
CONV
J.K. 1991
SEG
SEG
J.K. 1991
PC
PC
Koebe 1990
CIR
CIR
Bouchet 1985
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
21
Lower bound
Upper bound
STR
STR
J.K. 1991
CONV
CONV
J.K. 1991
J.K., Matoušek 1994
SEG
SEG
J.K. 1991
K-M 1994
PC
PC
Koebe 1990
CIR
CIR
Bouchet 1985
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
22
Lower bound
Upper bound
STR
STR
J.K. 1991
Pach, Tóth 2001 Schaefer, Štefankovic 2001
CONV
CONV
J.K. 1991
J.K., Matoušek 1994
SEG
SEG
J.K. 1991
K-M 1994
PC
PC
Koebe 1990
CIR
CIR
Bouchet 1985
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
23
Lower bound
Upper bound
STR
STR
J.K. 1991
Schaefer, Sedgwick, Štefankovic 2002
CONV
CONV
J.K. 1991
J.K., Matoušek 1994
SEG
SEG
J.K. 1991
K-M 1994
PC
PC
Koebe 1990
CIR
CIR
Bouchet 1985
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
24
Lower bound
Upper bound
STR
STR
J.K. 1991
Schaefer, Sedgwick, Štefankovic 2002
?
CONV
CONV
J.K. 1991
J.K., Matoušek 1994
?
SEG
SEG
J.K. 1991
K-M 1994
PC
PC
Koebe 1990
CIR
CIR
Bouchet 1985
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
25
Lower bound
Upper bound
STR
STR
J.K. 1991
Schaefer, Sedgwick, Štefankovic 2002
?
CONV
CONV
J.K. 1991
J.K., Matoušek 1994
?
SEG
SEG
J.K. 1991
K-M 1994
?
PC
PC
CIR
CIR
Bouchet 1985
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
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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 ? ?
28
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)
uv ? EG ? Cu ? Cv ?
separating lines
33
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
Cu
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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Lower bound
Upper bound
STR
STR
J.K. 1991
Schaefer, Sedgwick, Štefankovic 2002
?
CONV
CONV
J.K. 1991
J.K., Matoušek 1994
?
SEG
SEG
J.K. 1991
K-M 1994
?
PC
PC
CIR
CIR
Bouchet 1985
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
36
Polygon-circle graphs representable by polygons
of bounded size
37
Polygon-circle graphs representable by polygons
of bounded size
k-PC Intersection graphs of convex k-gons
inscribed to a circle
3-PC
2-PC CIR
4-PC
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Polygon-circle graphs representable by polygons
of bounded size
k-PC Intersection graphs of convex k-gons
inscribed to a circle
3-PC
2-PC CIR
4-PC
PC ? k-PC
?
k2
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Example forcing large number of corners
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Example forcing large number of corners
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Example forcing large number of corners
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PC
5-PC
4-PC
3-PC
CIR 2-PC
43
?
PC
5-PC
J.K., M. Pergel 2003
4-PC
3-PC
CIR 2-PC
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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

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• 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

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u1, u2, u3, u4, u5, u6
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u1, u2, u3, u4, u5, u6
ae, e ? E
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u1, u2, u3, u4, u5, u6
ae, e ? E
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u1, u2, u3, u4, u5, u6
ae, e ? E
bv, v ? V
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u1, u2, u3, u4, u5, u6
ae, e ? E
bv, v ? V
?(G) 3 ? H ? 3-PC
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u1, u2, u3, u4, u5, u6
ae, e ? E
bv, v ? V
?(G) gt 3 ? H ? 3-PC
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u1, u2, u3, u4, u5, u6
ae, e ? E
bv, v ? V
?(G) gt 3 ? H ? 3-PC
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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

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STRING graphs
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STRING graphs
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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., Matoušek 1991)
• At(n) ? 2cn
• Thm (Schaefer, Štefankovic 2001)
• At(n) ? n2n-2

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Sizes of SEG representations

• Rational endpoints of segments
• Integral endpoints
• Size of representation max coordinate of
  endpoint (in absolute value)

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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
• 22n

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Thm (Schaefer, Štefankovic 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.

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e crossed by e1, e2, , ek
e
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e crossed by e1, e2, , ek
e
(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
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e crossed by e1, e2, , ek
e
(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
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e crossed by e1, e2, , ek
e
(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 this vectors are the same, and
hence we find a segment on e where all
other edges have even number of
crossing points
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e
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e
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e
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2m crossing points with e 4m crossing points with
the circle
e
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2m crossing points with e 4m crossing points with
the circle
e
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2m crossing points with e 4m crossing points with
the circle
e
Circle inversion
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2m crossing points with e 4m crossing points with
the circle
e
Circle inversion
Symmetric flip
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2m crossing points with e 4m crossing points with
the circle
e
Circle inversion
Symmetric flip
2m crossing points with the circle, no
new crossing points arouse
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2m crossing points with e 4m crossing points with
the circle
e
Circle inversion
Symmetric flip
2m crossing points with the circle, no
new crossing points arouse
Reroute e along the semicircle with fewer number
of crossing points
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2m crossing points with e 4m crossing points with
the circle
e
Circle inversion
Symmetric flip
2m crossing points with the circle, no
new crossing points arouse
Reroute e along the semicircle with fewer number
of crossing points
Better realization - m lt 2m
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4. Optimization problems
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Determining the chromatic number
STR
CONV
SEG
PC
CIR
CA
INT
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?(G) ? k for fixed k
STR
CONV
SEG
PC
CIR
CA
INT
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Determining the independence number
STR
CONV
SEG
PC
CIR
CA
INT
82
Determining the clique number
STR
CONV
J.K., Nešetril 1989
SEG
PC
CIR
CA
INT
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Determining the independence number - Interval
filament graphs
STR
CONV
IFA
Gavril 2000
SEG
PC
CIR
CA
INT
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Interval filament graphs
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A-mixed graphs

• A is a class of graphs.
• G (V,E) is A-mixed if
• E E1 ? E2 and E2 is transitively oriented
  so that
• xy ? E2 and yz ? E1 imply xz ? E1 , and
• (V,E1) ? A

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Mixed condition
?
87

• Thm (Gavril 2000) If WEIGHTED CLIQUE is
  polynomial in graphs from class A, then it is
  also polynomial in A-mixed graphs.

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• Thm (Gavril 2000) If WEIGHTED CLIQUE is
  polynomial in graphs from class A, then it is
  also polynomial in A-mixed graphs.
• Thm (Gavril 2000)
• CO-IFA (CO-INT)-mixed

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• Thm (Gavril 2000) If WEIGHTED CLIQUE is
  polynomial in graphs from class A, then it is
  also polynomial in A-mixed graphs.
• Thm (Gavril 2000)
• CO-IFA (CO-INT)-mixed
• Corollary WEIGHTED INDEPENDENT SET is polynomial
  in IFA graphs

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Interval filament graphs
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Lower bound
Upper bound
STR
STR
J.K. 1991
Schaefer, Sedgwick, Štefankovic 2002
?
IFA
IFA
?
PC
PC
CIR
CIR
Bouchet 1985
CA
CA
Tucker 1970
INT
INT
Gilmore, Hoffman 1964
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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

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6. Representations of Co- Planar Graphs

• J.K., Kubena (1999) Co-Planar ? CONV
• Corollary CLIQUE is NP-hard for CONV graphs
• Problem Co-Planar ? SEG ?

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Thank you
95
6th International Czech-Slovak Symposium
onCombinatorics, Graph Theory, Algorithms and
ApplicationsPrague, July 10-15, 2006

• Honoring the 60th birthday of Jarik Nešetril