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Discrete & Computational Geometry

, Volume 19, Issue 3, pp 461–469 | Cite as

On Geometric Graphs with No k Pairwise Parallel Edges

  • P. Valtr

Abstract.

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>

Keywords

General Position Geometric Graph Parallel Edge Convex Quadrilateral 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Copyright information

© Springer-Verlag New York Inc. 1998

Authors and Affiliations

  • P. Valtr
    • 1
  1. 1.Department of Applied Mathematics, Charles University, Malostranské nám. 25, 118 00 Praha 1, Czech Republic valtr@kam.ms.mff.cuni.cz and DIMACS Center, Rutgers University, P.O. Box 1179, Piscataway, NJ 08855, USA valtr@dimacs.rutgers.eduUS

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