Recognizing String Graphs Is Decidable

  • János Pach
  • Géza Tóth
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)


A graph is called a string graph if its vertices can be represented by continuous curves (“strings”)in the plane so that two of them cross each other if and only if the corresponding vertices are adjacent. It is shown that there exists a recursive function f(n)with the property that every string graph of n vertices has a representation in which any two curves cross at most f(n)times. We obtain as a corollary that there is an algorithm for deciding whether a given graph is a string graph. This solves an old problem of (1959), (1966), and G(1971).


Intersection Graph Combinatorial Theory Permutation Graph Clockwise Order Delete Vertex 
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  1. [B59]
    S. Benzer: On the topology of the genetic fine structure, Proceedings of the National Academy of Sciences of the United States of America 45 (1959), 1607–1620.Google Scholar
  2. [CGP98a]
    Z.-Z. Chen, M. Grigni, and C. H. Papadimitriou, Planar map graphs, in: STOC’ 98, ACM, 1998, 514–523.Google Scholar
  3. [CGP98b]
    Z.-Z. Chen, M. Grigni, and C. H. Papadimitriou, Planar topological inference, (Japanese)in: Algorithms and Theory of Computing (Kyoto,1998) Sūrikaisekikenkyūsho Kōkyūroku 1041 (1998), 1–8.zbMATHMathSciNetGoogle Scholar
  4. [CHK99]
    Z.-Z. Chen, X. He, and M.-Y. Kao, Nonplanar topological inference and political-map graphs, in: Proceedings of the Tenth Annual ACM-SIAM Symposium on Discrete Algorithms (Baltimore,MD,1999), ACM, New York, 1999, 195–204.Google Scholar
  5. [DETT99]
    G. Di Battista, P. Eades, R. Tamassia, and I. G. Tollis, Graph Drawing, Prentice Hall, Upper Saddle River, NJ, 1999.zbMATHCrossRefGoogle Scholar
  6. [E93]
    M. Egenhofer, A model for detailed binary topological relationships, Geomatica 47 (1993), 261–273.Google Scholar
  7. [EF91]
    M. Egenhofer and R. Franzosa, Point-set topological spatial relations, International Journal of Geographical Information Systems 5 (1991), 161–174.CrossRefGoogle Scholar
  8. [ES93]
    M. Egenhofer and J. Sharma, Assessing the consistency of complete and incomplete topological information, Geographical Systems 1 (1993), 47–68.Google Scholar
  9. [EET76]
    G. Ehrlich, S. Even, and R. E. Tarjan, Intersection graphs of curves in the plane, Journal of Combinatorial Theory,Series B 21 (1976), 8–20.zbMATHCrossRefMathSciNetGoogle Scholar
  10. [EHP00]
    P. Erdős, A. Hajnal, and J. Pach, A Ramsey-type theorem for bipartite graphs, Geombinatorics 10 (2000), 64–68.MathSciNetGoogle Scholar
  11. [EPL72]
    S. Even, A. Pnueli, and A. Lempel, Permutation graphs and Transitive graphs, Journal of Association for Computing Machinery 19 (1972), 400–411.zbMATHMathSciNetGoogle Scholar
  12. [G80]
    M. C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Academic Press, New York, 1980.zbMATHGoogle Scholar
  13. [G78]
    R. L. Graham: Problem, in: Combinatorics,Vol. II (A. Hajnal and V. T. Sós, eds.), North-Holland Publishing Company, Amsterdam, 1978, 1195.Google Scholar
  14. [HT74]
    J. Hopcroft and R. E. Tarjan, Efficient planarity testing, J. ACM 21 (1974), 549–568.zbMATHCrossRefMathSciNetGoogle Scholar
  15. [K83]
    J. Kratochvíl, String graphs, in: Graphs and Other Combinatorial Topics (Prague,1982), Teubner-Texte Math. 59, Teubner, Leipzig, 1983, 168–172.Google Scholar
  16. [K91a]
    J. Kratochvíl, String graphs I: The number of critical nonstring graphs is infinite, Journal of Combinatorial Theory,Series B 52 (1991), 53–66.zbMATHCrossRefMathSciNetGoogle Scholar
  17. [K91b]
    J. Kratochvíl, String graphs II: Recognizing string graphs is NP-hard, Journal of Combinatorial Theory,Series B 52 (1991), 67–78.zbMATHCrossRefMathSciNetGoogle Scholar
  18. [K98]
    J. Kratochvíl, Crossing number of abstract topological graphs, in: Graph drawing (Montreal,QC,1998),Lecture Notes in Comput. Sci. 1547, Springer, Berlin, 1998, 238–245.CrossRefGoogle Scholar
  19. [KLN91]
    J. Kratochvíl, A. Lubiw, and J. Nešetřil, Noncrossing subgraphs in topological layouts, SIAM J. Discrete Math. 4 (1991), 223–244.zbMATHCrossRefMathSciNetGoogle Scholar
  20. [KM89]
    J. Kratochvíl and J. Matoušek, NP-hardness results for intersection graphs, Comment. Math. Univ. Carolin. 30 (1989), 761–773.zbMATHMathSciNetGoogle Scholar
  21. [KM91]
    J. Kratochvíl and J. Matoušek, String graphs requiring exponential representations, Journal of Combinatorial Theory,Series B 53 (1991), 1–4.zbMATHCrossRefMathSciNetGoogle Scholar
  22. [KM94]
    J. Kratochvíl and J. Matoušek, Intersection graphs of segments, Journal of Combinatorial Theory,Series B 62 (1994), 289–315.zbMATHCrossRefMathSciNetGoogle Scholar
  23. [LLR95]
    N. Linial, E. London, and Y. Rabinovich, The geometry of graphs and some of its algorithmic applications, Combinatorica 15 (1995), 215–245.zbMATHCrossRefMathSciNetGoogle Scholar
  24. [MP93]
    M. Middendorf and F. Pfeiffer, Weakly transitive orientations, Hasse diagrams and string graphs, in: Graph Theory and Combinatorics (Marseille-Luminy,1990),Discr ete Math. 111 (1993), 393–400.zbMATHMathSciNetGoogle Scholar
  25. [PS01]
    J. Pach and J. Solymosi, Crossing patterns of segments, Journal of Combinatorial Theory,Ser. A, to appear.Google Scholar
  26. [R99]
    S. Rao, Small distortion and volume preserving embeddings for planar and Euclidean metrics, in: Proceedings of the Fifteenth Annual Symposium on Computational Geometry (Miami Beach,FL,1999), ACM, New York, 1999, 300–306.Google Scholar
  27. [SS01]
    M. Schaefer and D. Stefankovič, Decidability of string graphs, STOC 01, to appear.Google Scholar
  28. [S66]
    F. W. Sinden, Topology of thin film RC circuits, Bell System Technological Journal (1966), 1639–1662.Google Scholar
  29. [SP92]
    T. R. Smith and K. K. Park, Algebraic approach to spatial reasoning, International Journal of Geographical Information Systems 6 (1992), 177–192.CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • János Pach
    • 1
  • Géza Tóth
    • 1
  1. 1.Rényi Institute of MathematicsHungarian Academy of SciencesHungary

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