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Geometric Representations of Graphs

Geometric Representations of Graphs. A survey of recent results and problems Jan Kratochvíl, Prague. Outline of the Talk. Intersection Graphs Recognition of the Classes Sizes of Representations Optimization Problems Interval Filament Graphs Representations of Planar Graphs.

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Geometric Representations of Graphs

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  1. Geometric Representationsof 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. Intervalgraphs INT

  5. Intervalgraphs INT Circular Arc graphs CA

  6. Intervalgraphs 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 Koebe 1990 PC PC 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 J.K. 1991 SEG Koebe 1990 PC PC CIR CIR Bouchet 1985 CA CA Tucker 1970 INT INT Gilmore, Hoffman 1964

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

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

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

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

  26. 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 ?

  27. Mv Mw Mu Mz {Mu, u  VG} uv  EG MuMv 

  28. Mv Xuv Xuw Mw Mu Xuz Mz ChooseXuv  MuMvfor every uv  EG

  29. Mv Xuv Xuw Mw Mu Xuz Mz ReplaceMuby Cu = conv(Xuv: vs.t. uv EG)  Mu CuCv  MuMv  uv  EG

  30. Introduce variables xuv , yuv  R s.t. Xuv = [xuv , yuv ] for uv EG

  31. Introduce variables xuv , yuv  R s.t. Xuv = [xuv , yuv ] for uv EG uv  EG  CuCv  guaranteed by the choiceCu = conv(Xuv: vs.t. uv EG)

  32. Introduce variables xuv , yuv  R s.t. Xuv = [xuv , yuv ] for uv EG uv  EG  CuCv  guaranteed by the choiceCu = conv(Xuv: vs.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 choiceCu = conv(Xuv: vs.t. uv EG) uw EG  CuCw= separating lines Cw Cu auwx + buwy + cuw = 0

  34. Introduce variables xuv , yuv  R s.t. Xuv = [xuv , yuv ] for uv EG uv  EG  CuCv  guaranteed by the choiceCu = conv(Xuv: vs.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) < 0 for all u,v,w,z s.t. uv, wz EG and uw EG

  35. Lower bound Upper bound Schaefer, Sedgwick, Štefankovič 2002 STR STR J.K. 1991 ? CONV J.K., Matoušek 1994 CONV J.K. 1991 ? SEG K-M 1994 J.K. 1991 SEG ? 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

  38. 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  k=2

  39. Example forcing large number of corners

  40. Example forcing large number of corners

  41. Example forcing large number of corners

  42. 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

  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 = {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}

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

  47. {u1,u2,u3,u4,u5,u6} {ae, e  E}

  48. {u1,u2,u3,u4,u5,u6} {ae, e  E}

  49. {u1,u2,u3,u4,u5,u6} {ae, e  E} {bv, v  V}

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

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