On Geometric Graphs with No k Pairwise Parallel Edges
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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 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. <lsiheader> <onlinepub>26 June, 1998 <editor>Editors-in-Chief: &lsilt;a href=../edboard.html#chiefs&lsigt;Jacob E. Goodman, Richard Pollack&lsilt;/a&lsigt; <pdfname>19n3p461.pdf <pdfexist>yes <htmlexist>no <htmlfexist>no <texexist>yes <sectionname> </lsiheader>