How Many Ways Can One Draw a Graph?

  • János Pach
  • Géza Tóth
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2912)


Using results from extremal graph theory, we determine the asymptotic number of string graphs with n vertices, i.e., graphs that can be obtained as the intersection graph of a system of continuous arcs in the plane. The number becomes much smaller, for any fixed d, if we restrict our attention to systems of arcs, any two of which cross at most d times. As an application, we estimate the number of different drawings of the complete graph K n with n vertices under various side conditions.


  1. [B59]
    Benzer, S.: On the topology of the genetic fine structure. Proceedings of the National Academy of Sciences of the United States of America 45, 1607–1620 (1959)CrossRefGoogle Scholar
  2. [B78]
    Bollobás, B.: Extremal Graph Theory, London Mathematical Society Monographs, vol. 11. Academic Press/Harcourt Brace Jovanovich, Publishers, London-New York (1978)Google Scholar
  3. [BT97]
    Bollobás, B., Thomason, A.: Hereditary and monotone properties of graphs. In: Graham, R.L., Nešetřil, J. (eds.) The mathematics of Paul Erdős II. Algorithms Combin, vol. 14, pp. 70–78. Springer, Berlin (1997)Google Scholar
  4. [CGP98a]
    Chen, Z.-Z., Grigni, M., Papadimitriou, C.H.: Planar map graphs. In: STOC 1998, pp. 514–523. ACM, New York (1998)CrossRefGoogle Scholar
  5. [CGP98b]
    Chen, Z.-Z., Grigni, M., Papadimitriou, C.H.: Planar topological inference. In: Algorithms and Theory of Computing, Kyoto. Sūrikaisekikenkyūsho Kōkyūroku, vol. 1041, pp. 1–8 (1998)Google Scholar
  6. [CHK99]
    Chen, Z.-Z., He, X., Kao, M.-Y.: Nonplanar topological inference and political-map graphs. In: Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms, Baltimore, MD, pp. 195–204. ACM, New York (1999)Google Scholar
  7. [Ch34]
    Chojnacki, C., Hanani, A.: Über wesentlich unplättbare Kurven im dreidimensionalen Raume. Fund. Math. 23, 135–142 (1934)Google Scholar
  8. [DETT99]
    Di Battista, G., Eades, P., Tamassia, R., Tollis, I.G.: Graph Drawing. Prentice Hall, Upper Saddle River (1999)MATHGoogle Scholar
  9. [E93]
    Egenhofer, M.: A model for detailed binary topological relationships. Geomatica 47, 261–273 (1993)Google Scholar
  10. [E65]
    Erdős, P.: On some extremal problems in extremal graph theory. Israel J. Math. 3, 113–116 (1965)CrossRefMathSciNetGoogle Scholar
  11. [EF91]
    Egenhofer, M., Franzosa, R.: Point-set topological spatial relations. International Journal of Geographical Information Systems 5, 161–174 (1991)CrossRefGoogle Scholar
  12. [ES93]
    Egenhofer, M., Sharma, J.: Assessing the consistency of complete and incomplete topological information. Geographical Systems 1, 47–68 (1993)Google Scholar
  13. [EET76]
    Ehrlich, G., Even, S., Tarjan, R.E.: Intersection graphs of curves in the plane. Journal of Combinatorial Theory, Series B 21, 8–20 (1976)MATHCrossRefMathSciNetGoogle Scholar
  14. [EFR86]
    Erdős, P., Frankl, P., Rödl, V.: The asymptotic number of graphs not containing a fixed subgraph and a problem for hypergraphs having no exponent. Graphs and Combinatorics 2, 113–121 (1986)CrossRefMathSciNetGoogle Scholar
  15. [ES66]
    Erdős, P., Simonovits, M.: A limit theorem in graph theory. Studia Sci. Math. Hungar. 1, 51–57 (1966)MathSciNetGoogle Scholar
  16. [ES46]
    Erdős, P., Stone, A.H.: On the structure of linear graphs. Bulletin Amer. Math. Soc. 52, 1087–1091 (1946)CrossRefGoogle Scholar
  17. [EPL72]
    Even, S., Pnueli, A., Lempel, A.: Permutation graphs and transitive graphs. Journal of Association for Computing Machinery 19, 400–411 (1972)MATHMathSciNetGoogle Scholar
  18. [G80]
    Golumbic, M.C.: Algorithmic Graph Theory and Perfect Graphs. Academic Press, New York (1980)MATHGoogle Scholar
  19. [GP86]
    Goodman, J.E., Pollack, R.: Upper bounds for configurations and polytopes in R d. Discrete Comput. Geom. 1, 219–227 (1986)MATHCrossRefMathSciNetGoogle Scholar
  20. [G78]
    Graham, R.L.: Problem. In: Hajnal, A., Sós, V.T. (eds.) Combinatorics, vol. II, p. 1195. North-Holland Publishing Company, Amsterdam (1978)Google Scholar
  21. [HT74]
    Hopcroft, J., Tarjan, R.E.: Efficient planarity testing. J. ACM 21, 549–568 (1974)MATHCrossRefMathSciNetGoogle Scholar
  22. [K83]
    Kratochvíl, J.: String graphs. In: Graphs and Other Combinatorial Topics (Prague, 1982), Teubner-Texte Math., Teubner, Leipzig, vol. 59, pp. 168–172 (1983)Google Scholar
  23. [K91a]
    Kratochvíl, J.: String graphs I: The number of critical nonstring graphs is infinite. Journal of Combinatorial Theory, Series B 52, 53–66 (1991)MATHCrossRefMathSciNetGoogle Scholar
  24. [K91b]
    Kratochvíl, J.: String graphs II: Recognizing string graphs is NP-hard. Journal of Combinatorial Theory, Series B 52, 67–78 (1991)MATHCrossRefMathSciNetGoogle Scholar
  25. [K98]
    Kratochvíl, J.: Crossing number of abstract topological graphs. In: Whitesides, S.H. (ed.) GD 1998. LNCS, vol. 1547, pp. 238–245. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  26. [PA95]
    Pach, J., Agarwal, P.K.: Combinatorial Geometry. Wiley Interscience, New York (1995)MATHGoogle Scholar
  27. [PS01]
    Pach, J., Solymosi, J.: Crossing patterns of segments. Journal of Combinatorial Theory, Ser. A 96, 316–325 (2001)MATHCrossRefMathSciNetGoogle Scholar
  28. [PT02]
    Pach, J., Tóth, G.: Recognizing string graphs is decidable. In: Mutzel, P., Jünger, M., Leipert, S. (eds.) GD 2001. LNCS, vol. 2265, pp. 247–260. Springer, Heidelberg (2002); Also in: Discrete and Computational Geometry 28, 593–606 (2002)CrossRefGoogle Scholar
  29. [PS92]
    Prömel, H.J., Steger, A.: Excluding induced subgraphs III: A general asymptotic. Random Structures and Algorithms 3, 19–31 (1992)MATHCrossRefMathSciNetGoogle Scholar
  30. [SS01a]
    Schaefer, M., Stefankovic̆, D.: Decidability of string graphs. In: Proceedings of the 33rd Annual Symposium on the Theory of Computing, pp. 241–246 (2001)Google Scholar
  31. [SS01b]
    Schaefer, M., Sedgwick, E., Stefankovic̆, D.: Recognizing string graphs in NP. In: Proceedings of the 34rd Annual Symposium on the Theory of Computing, pp. 1–6 (2002)Google Scholar
  32. [S66]
    Sinden, F.W.: Topology of thin film RC circuits. Bell System Technological Journal, 1639–1662 (1966)Google Scholar
  33. [SP92]
    Smith, T.R., Park, K.K.: Algebraic approach to spatial reasoning. International Journal of Geographical Information Systems 6, 177–192 (1992)CrossRefGoogle Scholar
  34. [T70]
    Tutte, W.T.: Toward a theory of crossing numbers. J. Combinatorial Theory 8, 45–53 (1970)MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • János Pach
    • 1
  • Géza Tóth
    • 1
  1. 1.Rényi InstituteHungarian Academy of SciencesBudapestHungary

Personalised recommendations