Triangle-Free Planar Graphs as Segments Intersection Graphs

  • N. de Castro
  • F. J. Cobos
  • J. C. Dana
  • A. Márquez
  • M. Noy
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1731)

Abstract

We prove that every triangle-free planar graph is the graph of intersection of a set of segments in the plane. Moreover, the segments can be chosen in only three directions (horizontal, vertical and oblique) and in such a way that no two segments cross, i.e., intersect in a common interior point.

Keywords

Planar Graph Intersection Graph Vertical Segment Outer Face Horizontal Segment 
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.
    D. W. Barnette, “On Steinitz’s theorem concerning convex 3-polytopes and on some properties of planar graphs”, The many facets of graph theory. Lectures Notes in Mathematics Vol. 110, Springer, Berlin, pp. 27–39, 1969.CrossRefGoogle Scholar
  2. 2.
    I. Ben-Arroyo Hartman, I. Newman and R. Ziv, “ On grid intersection graphs ”, Discrete Math. Vol. 87, pp. 41–52, 1991.MATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    H. de Fraysseix, P. Osona de Mendez and J. Pach, “Representation of planar graphs by segments”, Colloquia Mathematica Societatis Jáanos Bolyai, Intuitive Geometry, Szeged (Hungary), 1991.Google Scholar
  4. 4.
    M.C. Golumbic, “Algorithmic Graph Theory and Perfect Graphs”, Academic Press, 1980.Google Scholar
  5. 5.
    J. Kratochvíl, “A special planar satisfiability problem and a consequence of its NP-completeness”, Discrete Applied Math., Vol. 52, pp. 233–252, 1994.MATHCrossRefGoogle Scholar
  6. 6.
    W. Naji, “Reconnaisance des graphes de cordes”, Discrete Math. Vol. 54, pp. 329–337, 1985.MATHCrossRefMathSciNetGoogle Scholar
  7. 7.
    E. R. Scheinerman, “Intersection classes and multiple intersection parameters of graphs”, Ph D. thesis,, Princeton University, 1984.Google Scholar
  8. 8.
    J. Spinrad, “Recognition of circle graphs”, J. of Algorithms, Vol. 16, pp. 264–282, 1994.MATHCrossRefMathSciNetGoogle Scholar
  9. 9.
    C. Thomassen, “Grötzsch’s 3-colour theorem and its counterparts for the torus and the projective plane”, J. Comb. Theory B Vol. 62, pp. 268–279, 1994.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • N. de Castro
    • 1
  • F. J. Cobos
    • 1
  • J. C. Dana
    • 1
  • A. Márquez
    • 1
  • M. Noy
    • 2
  1. 1.Departamento de Matemática Aplicada IUniversidad de SevillaSpain
  2. 2.Dpto. de Matemática Aplicada IIUniversitat Politècnica de CatalunyaSpain

Personalised recommendations