On Geometric Graphs with No k Pairwise Parallel Edges

P. Valtr
1998 Discrete & Computational Geometry  
A geometric graph is a graph G = (V, E) drawn in the plane so that the vertex set V consists of points in general position and the edge set E consists of straight-line segments between points of V . Two edges of a geometric graph are said to be parallel if they are opposite sides of a convex quadrilateral. In this paper we show that, for any fixed k ≥ 3, any geometric graph on n vertices with no k pairwise parallel edges contains at most O(n) edges, and any geometric graph on n vertices with no
more » ... k pairwise crossing edges contains at most O(n log n) edges. We also prove a conjecture by Kupitz that any geometric graph on n vertices with no pair of parallel edges contains at most 2n − 2 edges.
doi:10.1007/pl00009364 fatcat:fzrmoqehj5hlhkzqjzsipm24ny