On Intersection Graphs of Segments with Prescribed Slopes

  • Jakub Černý
  • Daniel Král
  • Helena Nyklová
  • Ondřej Pangrác
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2265)

Abstract

We study intersection graphs of segments with prescribed slopes in the plane. A sufficient and necessary condition on tuples of slopes in order to define the same class of graphs is presented for both the possibilities that the parallel segments can or cannot overlap. Classes of intersection graphs of segments with four slopes are fully described; in particular, we find an infinite set of quadruples of slopes which define mutually distinct classes of intersection graphs of segments with those slopes.

Keywords

Distinct Classis Bold Line Geometric Object Intersection Graph Intersection Realization 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    T. Asano: Difficulty of the maximum independent set problem on intersection graphs of geometric objects, Graph theory, combinatorics and applications, vol.1, Wiley-Intersci.Publ., 1991, pp.9–18.MathSciNetGoogle Scholar
  2. 2.
    A. Bouchet: Reducing prime graphs and recognizing circle graphs, Combinatorica 7, 1987, pp.243–254.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    N.de Castro, F.J. Cobos, J.C. Dana, A. Marquez, M. Noy: Triangle-free planar graphs as segments intersection graphs, J. Krato chvil (ed.), Graph drawing, 7th international symposium, Štiřýn Castle, Czech Republic, proceedings, Springer LNCS 1731, 1999, pp.341–350.CrossRefGoogle Scholar
  4. 4.
    G. Ehrlich, S. Even, R.E. Tarjan: Intersection graphs of curves in the plane, J. Combinatorial Theory Ser.B 21, 1976, no.1, 8–20.MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    J. C. Fournier: Une caracterization des graphes de cordes, C.R. Acad. Sci. Paris 286A, 1978, pp.811–813.MathSciNetGoogle Scholar
  6. 6.
    H. de Fraysseix: A characterization of circle graphs, European Journal of Combinatorics 5, 1984, pp.223–238.MATHMathSciNetGoogle Scholar
  7. 7.
    H.de Fraysseix, P. Ossona de Mendez, J. Pach: Representation of planar graphs by segments, Intuitive Geometry 63, 1991, pp.109–117.Google Scholar
  8. 8.
    M. Goljan, J. Kratochvíl, P. Kučera: String graphs, Academia, Prague 1986.Google Scholar
  9. 9.
    I. B.-A. Hartman, I. Newman, R. Ziv: On grid intersection graphs, Discrete Math. 87, 1991, no.1, pp.41–52.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    V. B. Kalinin: On intersection graphs, Algorithmic constructions and their efficiency (in Russian), Yaroslav. Gos. Univ., 1983, pp.72–76.Google Scholar
  11. 11.
    S. Klavžar, M. Petkovšek: Intersection graphs of halflines and halfplanes, Discrete Math. 66, 1987, no.1–2, pp.133–137.CrossRefMATHMathSciNetGoogle Scholar
  12. 12.
    J. Kratochvíl: personal comunication.Google Scholar
  13. 13.
    J. Kratochvíl, J. Matoušek: Intersection Graphs of Segments, Journal of Combinatorial Theory, Series B, Vol.62, No.2, 1994, pp.289–315.MATHCrossRefGoogle Scholar
  14. 14.
    J. Kratochvíl, J. Matoušek: NP-hardness results for intersection graphs, Comment. Math.Univ.Carolin. 30, 1989, pp.761–773.MATHMathSciNetGoogle Scholar
  15. 15.
    J. Kratochvíl, J. Nešetřil: Independent set and clique problems in intersection defined classes of graphs, Comment.Math.Univ. Carolin. 31, 1990, pp.85–93.MATHMathSciNetGoogle Scholar
  16. 16.
    A. C. Tucker: An algorithm for circular-arc graphs, SIAM J.Computing 31.2, 1980, pp.211–216.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Jakub Černý
    • 1
  • Daniel Král
    • 1
  • Helena Nyklová
    • 1
  • Ondřej Pangrác
    • 1
  1. 1.Department of Applied Mathematics and Institute for Theoretical Computer Science (ITI)Charles UniversityPragueCzech Republic

Personalised recommendations