Advertisement

On Mutually Avoiding Sets

  • Pavel Valtr
Part of the Algorithms and Combinatorics book series (AC, volume 14)

Summary

Two finite sets of points in the plane are called mutually avoiding if any straight line passing through two points of anyone of these two sets does not intersect the convex hull of the other set. For any integer n, we construct a set of n points in general position in the plane which contains no pair of mutually avoiding sets of size more than \(O\left( {\sqrt n } \right)\) each. The given bound is tight up to a constant factor, since Aronov et al. [1] showed a polynomial-time algorithm for finding two mutually avoiding sets of size \(\Omega \left( {\sqrt n } \right)\) each in any set of n points in general position in the plane.

Keywords

Line Segment Convex Hull General Position Geometric Graph Convex Quadrilateral 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    B. Aronov, P. Erdős, W. Goddard, D. J. Kleitman, M. Klugerman, J. Pach, and L. J. Schulman, Crossing Families, Combinatorica 14 (1994), 127–134; also Proc. Seventh Annual Sympos. on Comp. Geom., ACM Press, New York (1991), 351–356.MathSciNetMATHCrossRefGoogle Scholar
  2. 2.
    I. Bárány, personal communication.Google Scholar
  3. 3.
    P. Erdős, On some problems of elementary and combinatorial geometry, Ann. Mat. Pura. Appi. (4) 103 (1975), 99–108.CrossRefGoogle Scholar
  4. 4.
    P. Erdős and G. Szekeres, A combinatorial problem in geometry, Compositio Math. 2 (1935), 463–470.MathSciNetGoogle Scholar
  5. 5.
    L. Fejes Tóth, Regular figures, Pergamon Press, Oxford 1964.MATHGoogle Scholar
  6. 6.
    H. Harborth, Konvexe Fünfecke in ebenen Punktmengen, Elem. Math. 33 (1978), 116–118.MathSciNetMATHGoogle Scholar
  7. 7.
    J. D. Horton, Sets with no empty convex 7-gons, Canadian Math. Bull. 26 (1983), 482–484.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    J. Pach, Notes on Geometric Graph Theory, DIMACS Series in Discrete Mathematics and Theoretical Computer Science, Vol. 6 (1991), 273–285.Google Scholar
  9. 9.
    P. Valtr, Convex independent sets and 7-holes in restricted planar point sets, Discrete Comput. Geom. 7 (1992), 135–152.MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Pavel Valtr
    • 1
    • 2
  1. 1.Department of Applied MathematicsCharles UniversityCzech Republic
  2. 2.Graduiertenkolleg “Algorithmische Diskrete Mathematik”, Fachbereich MathematikFreie Universität BerlinBerlinGermany

Personalised recommendations