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On Geometric Graphs with No k Pairwise Parallel Edges

1998
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Discrete & Computational Geometry
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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

doi:10.1007/pl00009364
fatcat:fzrmoqehj5hlhkzqjzsipm24ny