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Algorithmica

, Volume 16, Issue 1, pp 111–117 | Cite as

Applications of the crossing number

  • J. Pach
  • F. Shahrokhi
  • M. Szegedy
Article

Abstract

LetG be a graph ofn vertices that can be drawn in the plane by straight-line segments so that nok+1 of them are pairwise crossing. We show thatG has at mostc k nlog2k−2n edges. This gives a partial answer to a dual version of a well-known problem of Avital-Hanani, Erdós, Kupitz, Perles, and others. We also construct two point sets {p1,⋯,p n }, {q1,⋯,q n } in the plane such that any piecewise linear one-to-one mappingfR2R2 withf(pi)=qi (1≤in) is composed of at least Ω(n2) linear pieces. It follows from a recent result of Souvaine and Wenger that this bound is asymptotically tight. Both proofs are based on a relation between the crossing number and the bisection width of a graph.

Key words

Crossing number Geometric graph Bisection width Triangulation 

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Copyright information

© Springer-Verlag New York Inc. 1996

Authors and Affiliations

  • J. Pach
    • 1
    • 2
  • F. Shahrokhi
    • 3
  • M. Szegedy
    • 4
  1. 1.Department of Computer Science, City CollegeCUNYNew YorkUSA
  2. 2.Courant InstituteNYUNew YorkUSA
  3. 3.Department of Computer ScienceUniversity of North TexasDentonUSA
  4. 4.AT&T Bell LaboratoriesMurray HillUSA

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