On Mutually Avoiding Sets

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


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.


Line Segment Convex Hull General Position Geometric Graph Convex Quadrilateral 
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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

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