On s-intersecting curves and related problems

Sarit Buzaglo, Rom Holzman, Rom Pinchasi
2008 Proceedings of the twenty-fourth annual symposium on Computational geometry - SCG '08  
Let P be a set of n points in the plane and let C be a family of simple closed curves in the plane each of which avoids the points of P . For every curve C ∈ C we denote by disc(C) the region in the plane bounded by C. Fix an integer k ≥ 0 and assume that every two curves in C intersect at most k times and that for every two curves C, C ′ ∈ C the intersection disc(C) ∩ disc(C ′ ) is a connected set. We consider the family F = {P ∩ disc(C) | C ∈ C}. When k is even, we provide sharp bounds, in
more » ... sharp bounds, in terms of n, k, and ℓ, for the number of sets in F of cardinality ℓ, assuming that ∩ C∈C disc(C) is nonempty. In particular, we provide sharp bounds for the number of halving pseudo-parabolas for a set of n points in the plane. Finally, we consider the VC-dimension of F and show that F has VC-dimension at most k + 1. √ log k) by Tóth ([10]).
doi:10.1145/1377676.1377690 dblp:conf/compgeom/BuzagloHP08 fatcat:umhfi4fom5frvmu2o3yvvgn3mi