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 {M u, u  V G } uv  E G  M u  M v  

4. Interval graphs INT

5. Circular Arc graphs CA

6. Interval graphs INT Circular Arc graphs CA Circle graphs CIR

7. Circular Arc graphs CA Circle graphs CIR Polygon-Circle graphs PC

8. SEG

9. CONV

10. SEG CONV STRING

11. INT CA CIR PC CONV STR SEG

12. 2. Complexity of Recognition Upper bound Lower bound P NP NP-hard PSPACE Decidable Unknown

13. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound

14. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound

15. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964

16. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964

17. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964 Tucker 1970

18. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964 Tucker 1970 Bouchet 1985

19. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964 Tucker 1970 Bouchet 1985 Koebe 1990

20. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964 Tucker 1970 Bouchet 1985 Koebe 1990 J.K. 1991

21. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964 Tucker 1970 Bouchet 1985 Koebe 1990 J.K. 1991 J.K., Matoušek 1994 K-M 1994

22. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964 Tucker 1970 Bouchet 1985 Koebe 1990 J.K. 1991 J.K., Matoušek 1994 K-M 1994 Pach, Tóth 2001; Schaefer, Štefankovič 2001

23. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964 Tucker 1970 Bouchet 1985 Koebe 1990 J.K. 1991 J.K., Matoušek 1994 K-M 1994 Schaefer, Sedgwick, Štefankovič 2002

24. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964 Tucker 1970 Bouchet 1985 Koebe 1990 J.K. 1991 J.K., Matoušek 1994 K-M 1994 Schaefer, Sedgwick, Štefankovič 2002 ? ?

25. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964 Tucker 1970 Bouchet 1985 J.K. 1991 J.K., Matoušek 1994 K-M 1994 Schaefer, Sedgwick, Štefankovič 2002 ? ? ?

26. Thm: Recognition of CONV graphs is in PSPACE  Reduction to solvability of polynomial inequalities in R:  x 1, x 2, x 3 … x n  R s.t. P 1 (x 1, x 2, x 3 … x n ) > 0 P 2 (x 1, x 2, x 3 … x n ) > 0 … P m (x 1, x 2, x 3 … x n ) > 0 ?

27. {M u, u  V G } uv  E G  M u  M v   MuMu MvMv MwMw MzMz

28. MuMu MvMv MwMw MzMz Choose X uv  M u  M v for every uv  E G X uw X uz X uv

29. C u  C v    M u  M v    uv  E G MuMu MvMv MwMw MzMz Replace M u by C u = conv(X uv : v s.t. uv  E G )  M u X uw X uz X uv

30. Introduce variables x uv, y uv  R s.t. X uv = [x uv, y uv ] for uv  E G

31. uv  E G  C u  C v   guaranteed by the choice C u = conv(X uv : v s.t. uv  E G )

32. Introduce variables x uv, y uv  R s.t. X uv = [x uv, y uv ] for uv  E G uv  E G  C u  C v   guaranteed by the choice C u = conv(X uv : v s.t. uv  E G ) uv  E G  C u  C v =  separating lines

33. Introduce variables x uv, y uv  R s.t. X uv = [x uv, y uv ] for uv  E G uv  E G  C u  C v   guaranteed by the choice C u = conv(X uv : v s.t. uv  E G ) uw  E G  C u  C w =  separating lines CuCu CwCw a uw x + b uw y + c uw = 0

34. Introduce variables x uv, y uv  R s.t. X uv = [x uv, y uv ] for uv  E G uv  E G  C u  C v   guaranteed by the choice C u = conv(X uv : v s.t. uv  E G ) uw  E G  C u  C w =  separating lines CuCu CwCw a uw x + b uw y + c uw = 0 Representation is described by inequalities (a uw x uv + b uw y uv + c uw ) (a uw x wz + b uw y wz + c uw ) < 0 for all u,v,w,z s.t. uv, wz  E G and uw  E G

35. INT CA CIR PC CONV STR SEG INT CA CIR PC CONV STR SEG Upper bound Lower bound Gilmore, Hoffman 1964 Tucker 1970 Bouchet 1985 J.K. 1991 J.K., Matoušek 1994 K-M 1994 Schaefer, Sedgwick, Štefankovič 2002 ? ? ?

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 2-PC = CIR 3-PC 4-PC

38. Polygon-circle graphs representable by polygons of bounded size k-PC = Intersection graphs of convex k-gons inscribed to a circle 2-PC = CIR 3-PC 4-PC PC =  k-PC  k=2

39. Example forcing large number of corners

42. 3-PC CIR = 2-PC PC 4-PC 5-PC

43. 3-PC CIR = 2-PC PC 4-PC 5-PC J.K., M. Pergel 2003 ?

44. 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

45. W = {u 1, u 2, u 3, u 4, u 5, u 6 }  {a e, e  E}  {b v, v  V} F = {u 1 u 2, u 2 u 3, u 3 u 4, u 4 u 5, u 5 u 6, u 6 u 1 }  {a e b v, v  e  E}  {b u b v, u,v  V}  {b v u i, v  V, i = 2,4,6}

46. {u1, u2, u3, u4, u5, u6}{u1, u2, u3, u4, u5, u6}

47. {u1, u2, u3, u4, u5, u6}{u1, u2, u3, u4, u5, u6} {a e, e  E}

48. {u1, u2, u3, u4, u5, u6}{u1, u2, u3, u4, u5, u6}

49. {u1, u2, u3, u4, u5, u6}{u1, u2, u3, u4, u5, u6} {b v, v  V}

50. {u1, u2, u3, u4, u5, u6}{u1, u2, u3, u4, u5, u6} {a e, e  E} {b v, v  V}  ’(G) = 3  H  3-PC

51. {u1, u2, u3, u4, u5, u6}{u1, u2, u3, u4, u5, u6} {a e, e  E} {b v, v  V}  ’(G) > 3  H  3-PC

52. {u1, u2, u3, u4, u5, u6}{u1, u2, u3, u4, u5, u6} {a e, e  E} {b v, v  V}  ’(G) > 3  H  3-PC

53. 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

54. STRING graphs

56. 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, D e  D f    ef  R  G = (V,E), R =  is realizable iff G is planar

59. 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

60. STRING graphs requiring large representations  Thm (J.K., Matoušek 1991): At(n)  2 cn  Thm (Schaefer, Štefankovič 2001): At(n)  n2 n-2

61. Sizes of SEG representations  Rational endpoints of segments  Integral endpoints  Size of representation = max coordinate of endpoint (in absolute value)

62. Sizes of SEG representations  Thm (J.K., Matoušek 1994) For every n, there is a SEG graph G n with O(n 2 ) vertices such that every SEG representation has size at least 2 2 n

63. Thm (Schaefer, Štefankovič 2001): At(n)  n2 n-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 2 k -1 crossing points.

64. e e crossed by e 1, e 2, …, e k

65. e (u 1, u 2, …, u k ) - binary vector expressing the parity of the number of intersections of e and e i between the beginning of e and this location

66. e e crossed by e 1, e 2, …, e k (u 1, u 2, …, u k ) - binary vector expressing the parity of the number of intersections of e and e i between the beginning of e and this location If the number of crossing points on e is  2 k, two of these vectors are the same

67. e e crossed by e 1, e 2, …, e k (u 1, u 2, …, u k ) - binary vector expressing the parity of the number of intersections of e and e i between the beginning of e and this location If the number of crossing points on e is  2 k, 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

68. e

69. e

70. e

71. e 2m crossing points with e 4m crossing points with the circle

72. e 2m crossing points with e 4m crossing points with the circle

73. e 2m crossing points with e 4m crossing points with the circle Circle inversion

74. e 2m crossing points with e 4m crossing points with the circle Circle inversion Symmetric flip

75. e 2m crossing points with e 4m crossing points with the circle Circle inversion Symmetric flip 2m crossing points with the circle, no new crossing points arouse

76. e 2m crossing points with e 4m crossing points with the circle 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

77. e 2m crossing points with e 4m crossing points with the circle 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 < 2m

78. 4. Optimization problems

79. INT CA CIR PC CONV STR SEG Determining the chromatic number

80. INT CA CIR PC CONV STR SEG  (G)  k for fixed k

81. INT CA CIR PC CONV STR SEG Determining the independence number

82. INT CA CIR PC CONV STR SEG Determining the clique number J.K., Nešetřil 1989

83. INT CA CIR PC CONV STR SEG Determining the independence number - Interval filament graphs IFA Gavril 2000

84. Interval filament graphs

85. A A-mixed graphs  A  A is a class of graphs. A  G = (V,E) is A-mixed if E = E 1  E 2 and E 2 is transitively oriented so that  xy  E 2 and yz  E 1 imply xz  E 1, and A  (V,E 1 )  A

86.  Mixed condition

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

88.  Thm (Gavril 2000): CO-IFA = (CO-INT)-mixed

89. A, A  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

90. Interval filament graphs

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

92. 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

93. 6. Representations of Co- Planar Graphs  J.K., Kuběna (1999): Co-Planar  CONV  Corollary: CLIQUE is NP-hard for CONV graphs  Problem: Co-Planar  SEG ?

94. Thank you

95. 6th International Czech-Slovak Symposium on Combinatorics, Graph Theory, Algorithms and Applications Prague, July 10-15, 2006 Honoring the 60th birthday of Jarik Nešetřil