Contact graphs of curves

Extended abstract
  • Petr Hliněný
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1027)


Contact graphs are a special kind of intersection graphs of geometrical objects in which we do not allow the objects to cross but only to touch each other. Contact graphs of simple curves (and line segments as a special case) in the plane are considered. Several classes of contact graphs are introduced and their properties and inclusions between them are studied. Also the relation between planar and contact graphs is mentioned. Finally, it is proved that the recognition of contact graphs of curves (line segments) is NP-complete (NP-hard) even for planar graphs.


  1. 1.
    K. S. Booth, G. S. Lucker, Testing for the consecutive ones property, interval graphs, and graph planarity using PQ-tree algorithms, J. Comp. Systems Sci. 13 (1976), 255–265.Google Scholar
  2. 2.
    A. Bouchet, Reducing prime graphs and recognizing circle graphs, Combinatorica 7 (1987), 243–254.Google Scholar
  3. 3.
    G. Ehrlich, S. Even, R.E. Tarjan, Intersection graphs of curves in the plane, J. of Comb. Theory Ser. B 21 (1976), 8–20.CrossRefGoogle Scholar
  4. 4.
    H. de Fraysseix, P.O. de Mendez, J. Pach, Representation of planar graphs by segments, 63. Intuitive Geometry (1991), 110–117.Google Scholar
  5. 5.
    H. de Fraysseix, P.O. de Mendez, P. Rosenstiehl, On triangle contact graphs, Combinatorics, Probability and Computing 3 (1994), 233–246.Google Scholar
  6. 6.
    H. de Fraysseix, P.O. de Mendez, to appear.Google Scholar
  7. 7.
    M.R. Garey, D.S. Johnson, Computers and Intractability, W.H. Freeman and Company, New York 1979.Google Scholar
  8. 8.
    F. Gavril, Algorithms for a maximum clique and maximum independent set of a circle graph, Networks 4 (1973), 261–273.Google Scholar
  9. 9.
    P. Hliněný, Contact graphs of curves, KAM Preprint Series 95-285, Dept. of Applied Math., Charles University, Czech rep., 1995.Google Scholar
  10. 10.
    P. Koebe, Kontaktprobleme der konformen Abbildung, Berichte über die Verhandlungen der Sächsischen, Akad. d. Wiss., Math.-Physische Klasse 88 (1936), 141–164.Google Scholar
  11. 11.
    J. Kratochvíl, String graphs II: Recognizing string graphs is NP-hard, J. of Comb. Theory Ser. B 1 (1991), 67–78.CrossRefGoogle Scholar
  12. 12.
    J. Kratochvíl, J. Matoušek, Intersection graphs of segments, J. of Comb. Theory Ser. B 2 (1994), 289–315.CrossRefGoogle Scholar
  13. 13.
    J. Kratochvíl, J. Matoušek, String graphs requiring exponential representations, J. of Comb. Theory Ser. B 2 (1991), 1–4.CrossRefGoogle Scholar
  14. 14.
    C.B. Lekkerkerker, J.C. Boland, Representation of finite graphs by a set of intervals on the real line, Fund. Math. 51 (1962), 45–64.Google Scholar
  15. 15.
    S. Roberts, On the boxicity and cubicity of a graph, in “Recent Progresses in Combinatorics” 301–310, Academic Press, New York, 1969.Google Scholar
  16. 16.
    F.W. Sinden, Topology of thin film RC-circuits, Bell System Techn. J. (1966), 1639–1662.Google Scholar
  17. 17.
    C. Thomassen, presentation at Graph Drawing '93, Paris, 1993.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1996

Authors and Affiliations

  • Petr Hliněný
    • 1
  1. 1.Dept. of Applied MathematicsCharles UniversityPraha 1Czech Republic

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