1. Geometric Representations of Graphs Jan Kratochvíl, DIMATIA, Prague

2. Intersection Graphs {M u, u  V G } uv  E G  M u  M v  

3. String Graphs {M u, u  V G } uv  E G  M u  M v  

4. Personal Recollections 1982 – Czech-Slovak Graph Theory, Prague

5. Personal Recollections 1982 – Czech-Slovak Graph Theory, Prague 1983 – Prague 1990 – Tempe, Arizona

6. Personal Recollections 1982 – Czech-Slovak Graph Theory, Prague 1983 – Prague 1990 – Tempe, Arizona 1988 – Bielefeld, Germany

7. Intersection Graphs Every graph is an intersection graph.

8. Intersection Graphs Every graph is an intersection graph. M u = {e  E G | u  e}

9. Intersection Graphs Every graph is an intersection graph. uv  E G  M u  M v   M u = {e  E G | u  e}

10. Intersection Graphs Every graph is an intersection graph Restricting the sets

11. Intersection Graphs Every graph is an intersection graph Restricting the sets – by geometrical shape Motivation and applications in scheduling, biology, VLSI designs …

12. 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

13. Few Examples

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

15. Few Examples Interval graphs Interval graphs - - neat characterization chordal + co-comparability - recognizble in linear time - most optimization problems solvable in polynomial time - perfect

16. Few Examples SEG graphs SEG graphs - Ehrlich, Even, Tarjan 1976 Scheinerman Erdös, Gyarfás 1987 JK, Nešetřil 1990 JK, Matoušek 1994 Thomassen 2002

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

18. 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.

19. Few Examples String graphs String graphs - Sinden 1966 Ehrlich, Even, Tarjan 1976 JK 1991 JK, Matoušek 1991 Pach, Tóth 2001 Štefankovič, Schaffer 2001, 2002

20. Few Examples CONV graphs CONV graphs - Ogden, Roberts 1970 JK, Matoušek 1994 Agarwal, Mustafa 2004 Kim, Kostochka, Nakprasit 2004

21. Few Examples PC graphs PC graphs - Fellows 1988 Koebe 1990 JK, Kostochka 1994 Spinrad JK, Pergel 2002 Pergel 2007

22. Few Examples Circle graphs Circle graphs - De Fraysseix 1984 Bouchet 1985 Gyarfas 1987 Unger 1988 Kloks 1993 Kostochka 1994

23. Few Examples Circle graphs Circle graphs - - recognizable in linear time - coloring NP-hard - independent set, clique solvable in polynomial time - near-perfect  log     O(2  ) - close bounds open

24. Few Examples Circular Arc graphs Circular Arc graphs - Tucker 1971, 1980 Gavril 1974 Gyarfás 1987 Spinrad 1988 Hell, Bang-Jensen, Huang 1990 …

25. Few Examples Circular Arc graphs Circular Arc graphs - Tucker 1971, 1980 Gavril 1974 Gyarfás 1987 Spinrad 1988 Hell, Bang-Jensen, Huang 1990 …

29. Outline  String graphs  CONV and PC graphs  Representations of planar graphs

30. 1. String graphs Sinden 1966

31. 1. String graphs Sinden 1966 = IG(regions)

32. 1. String graphs Sinden 1966 = IG(regions) Graham 1974

33. 1. String graphs Sinden 1966 JK, Goljan, Kučera 1982

34. 1. String graphs Sinden 1966 JK, Goljan, Kučera 1982 Thomas 1988 IG(topologically con) = all graphs, String = IG(arc-connected sets)

35. 1. String graphs Sinden 1966 JK, Goljan, Kučera 1982 Thomas 1988 JK 1991 – NP-hard

36. 1. String graphs SEG CONV STRING

37. 1. String graphs SEG CONV STRING

38. 1. String graphs Sinden 1966 JK, Goljan, Kucera 1982 Thomas 1988 JK 1991 – NP-hard Recognition in NP?

39. 1. String graphs Sinden 1966 JK, Goljan, Kucera 1982 Thomas 1988 JK 1991 – NP-hard Recognition in NP?

40. 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

41. 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

42. String graphs requiring large representations  Thm (J.K., Matoušek 1991): At(n)  2 cn

47. 1. String graphs Sinden 1966 JK, Goljan, Kucera 1982 Thomas 1988 JK 1991 – NP-hard Recognition in NP? Are they recognizable at all?

48. Thm (Pach, Tóth 2001): At(n)  n n Thm (Schaefer, Štefankovič 2001): At(n)  n2 n-2

49. 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)

50. 1. Some subclasses

52. Complements of Comparability graphs (Golumbic 1977)

53. Co-comparability graphs

55. =  

56. Co-comparability graphs

58. 1. Some subclasses “Zwischenring” graphs NP-hard (Middendorf, Pfeiffer)

59. 1. Some subclasses Outerstring graphs NP-hard (Middendorf, Pfeiffer)

60. 1. Some subclasses Outerstring graphs NP-hard (Middendorf, Pfeiffer)

61. 1. Some subclasses Interval filament graphs (Gavril 2000) CLIQUE and IND SET can be solved in polynomial time

62. 2. CONV and PC JK, Matoušek 1994 – recognition in PSPACE

63. 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 ?

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

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

66. 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

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

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

69. 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

70. 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

71. 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 X uv X wz

72. 2. Recognition – NP-membership “Guess and verify”

73. 2. Recognition – NP-membership “Guess and verify” -INT, CA, CIR, PC, Co-Comparability -IFA – mixing characterization - CONV, SEG ? !! String – Lempel-Ziv compression

74. 2. Recognition – NP-membership Thm (JK, Matoušek 1994): For every n there is a graph G n  SEG with O(n 2 ) vertices s.t. every SEG representation with integer endpoints has a coordinate of absolute value  2 2 n. Same for CONV (Pergel 2008).

75. 2. CLIQUE in CONV graphs -CO-PLANAR  CONV (JK, Kuběna 99)

82. 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)

83. 2. CLIQUE in MAX-TOL graphs

84. 2. MAX-TOLERANCE (Golumbic, Trenk 2004)

85. 2. MAX-TOLERANCE S R S = {I u | u  V G } intervals, t u  R tolerances uv  E G iff |I u  I v | ≥ max {t u, t v }

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

87. 2. MAX-TOLERANCE IuIu tutu TuTu IvIv TvTv

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

89. A B C

90. Maximal cliques Q a maximal clique

91. Maximal cliques h highest basis of Q, v rightmost vertical side, t lowest diagonal side Q a maximal clique t h v

92. Maximal cliques Q(h,v,t) = all triangles that intersect h,v and t Q a maximal clique t h v

93. Claim: Q(h,v,t) = Q

94. Proof: 1) Q  Q(h,v,t) h

95. Claim: Q(h,v,t) = Q Proof: 1) Q  Q(h,v,t) 2) Q(h,v,t) is a clique

96. 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.

97. Claim: Q(h,v,t) = Q Proof: 1) Q  Q(h,v,t) 2) Q(h,v,t) is a clique a b

98. 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

99. Maximal cliques Q(h,v,t) = all triangles that intersect h,v and t Hence G has O(n 3 ) maximal cliques. Q a maximal clique t h v

100. 2. Polygon-circle graphs PC graphs PC graphs - Fellows 1988 Koebe 1990 JK, Kostochka 1994 Spinrad JK, Pergel 2002 Pergel 2007

101. 2. Polygon-circle graphs PC graphs PC graphs - Fellows 1988 Koebe 1990 JK, Kostochka 1994 Spinrad JK, Pergel 2002 Pergel 2007

102. 2. Polygon-circle graphs PC graphs PC graphs - Fellows 1988 Koebe 1990 JK, Kostochka 1994 Spinrad JK, Pergel 2002 Pergel 2007

103. 2. Polygon-circle graphs CIR PC IFA CA CHOR

104. 2. Polygon-circle graphs CIR PC IFA CA CHOR

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

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

107. 2. Short cycles Do short cycles help?

108. 2. Short cycles Do short cycles mind? Does large girth help?

109. DISK UNIT-DISK

110. DISK UNIT-DISK PSEUDO-DISK

111. 2. Short cycles Thm (J.K. 1996) Triangle-free intersection graphs of pseudodisks are planar.

112. 2. Short cycles Thm (J.K. 1996) Triangle-free intersection graphs of pseudodisks are planar.

113. 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.

114. Koebe (1936)

117. 2. Short cycles Thm (J.K. 1996) Triangle-free STRING graphs are NP-hard to recognize.

118. 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.

119. 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.

120. 3. Representations of planar graphs

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

123. 3. Representations of planar graphs -Planar graphs are exactly contact graphs of disks (Koebe 1934) -PLANAR  DISK -PLANAR  CONV -PLANAR  2-STRING

127. 3. Representations of planar graphs -PLANAR  2-STRING -Problem (Fellows 1988): Planar  1-STRING ? -True: Chalopin, Gonçalves, and Ochem [SODA 2007]

128. 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, …)

129. 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.)

130. 3. Bipartite planar graphs De Fraysseix, Ossona de Mendez, Pach d c f e b a 1 2 35 6 4 7 a b c d e f 1234567

131. 3. Bipartite planar graphs De Fraysseix, Ossona de Mendez, Pach a b c d e f 1234567

132. 3. Bipartite planar graphs De Fraysseix, Ossona de Mendez, Pach a b c d e f 1234567

133. 3. Representations of planar graphs -PLANAR  CONV -Planar graphs are contact graphs of triangles (de Fraysseix, Ossona de Mendez 1997)

135. 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?

136. 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

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

138. 1 2 3 b c a 1 2 3 ab c

139. 1 2 3 b c a 1 2 3 ab c

140. 3. Planar – open problems -PLANAR  MAX-TOL? (Lehmann) (i.e. are planar graphs intersection graphs of homothetic triangles?)

141. 3. Planar – open problems -PLANAR  MAX-TOL? (Lehmann) -Conjecture (Felsner, JK 2007): Planar 4-connected triangulations are contact graphs of homothetic triangles.

142. 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.

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

144. 1 2 3 b c a ab c

145. 1 2 3 b c a ab c

146. 4. Invitation  Graph Drawing, Crete, Sept 21 – 24, 2008  Prague MCW, July 28 – Aug 1, 2008