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A characterization of planar graphs by pseudo-line arrangements

  • Hisao Tamaki
  • Takeshi Tokuyama
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Part of the Lecture Notes in Computer Science book series (LNCS, volume 1350)

Abstract

Let Γ be an arrangement of pseudo-lines, i.e., a collection of unbounded x-monotone curves in which each curve crosses each of the others exactly once. A pseudo-line graph (ΓT, E) is a graph for which the vertices are the pseudo-lines of Γ and the edges are some un-ordered pairs of pseudo-lines of Γ. A diamond of pseudo-line graph (Γ, E) is a pair of edges p, q), p′, q′) ∈ E, (p′, q′) ∩ p′, q′ = 0, such that the crossing point of the pseudo-lines p and q lies vertically between p′ and q′ and the crossing point of p′ and q′ lies vertically between p and q. We show that a graph is planar if and only if it is isomorphic to a diamond-free pseudo-line graph. An immediate consequence of this theorem is that the O(kn) upper bound on the k-level complexity of an arrangement of straight-lines, which is very recently discovered by Dey, holds for an arrangement of pseudo-lines as well.

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References

  1. 1.
    M. Ajtai, V. Chvátal, M.M. Newborn, and E. Szemerédi, Crossing-free subgraphs, Ann. Discrete Math. 12 (1982), 9–12.Google Scholar
  2. 2.
    P.K. Agarwal, B. Aronov, and M. Sharir, On Levels in Arrangements of Lines, Segments, Planes, and Triangles, Proc. 13th Symposium on Computational Geometry, 1997, 30–38.Google Scholar
  3. 3.
    T.K. Dey, Improved Bounds for k-sets and k-th Levels, manuscript, 1997.Google Scholar
  4. 4.
    P. Erdös, L. Lovász, A. Simmons, and E.G. Straus. Dissection graphs of planar point sets. In A Survey of Combinatorial Theory, J.N. Srivastava et al., eds., 1973, 139–149, North-HOlland, Amsterdam.Google Scholar
  5. 5.
    I. Fáry. On straight line representation of planar graphs. Acta Sci. Math. Szeged., 11, 229–33, 1948.Google Scholar
  6. 6.
    L. Lovász. On the number of halving lines. Ann. Univ. Sci. Budapest, Eötös, Soc. Math. 14 (1971), 107–108.Google Scholar
  7. 7.
    J. Matoušek, Combinatorial and algorithmic geometry, Unpublished lecture notes, 1996.Google Scholar
  8. 8.
    G.H. Meister, Polygons have ears, Amer. Math. Mon. 82, 1975, 648–651.Google Scholar
  9. 9.
    J. Pack and P.K. Agarwal, Combinatorial Geometry, John Wiley and Sons, 1995.Google Scholar
  10. 10.
    J. Pach, W. Steiger, and E. Szemerédi. An upper bound on the number of planar k-sets. In Proc. 30th Ann. IEEE Symposium on Foundation of Computer Science, 1989, 72–79.Google Scholar
  11. 11.
    H. Tamaki and T. Tokuyama, How to cut pseudo-parabolas into segments, In Proc. 11th Symposium on Computational Geometry, 1995, 230–237.Google Scholar
  12. 12.
    H. Tamaki and T. Tokuyama, A characterization of planar graphs by pseudo-line arrangements, manuscript, August 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Hisao Tamaki
    • 1
  • Takeshi Tokuyama
    • 2
  1. 1.Dept of Computer ScienceMeiji UniversityKwasakiJapan
  2. 2.IBM Tokyo Research LaboratoryYamatoJapan

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