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Planar Point Sets Determine Many Pairwise Crossing Segments

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János Pach, Gábor Tardos
2019
conference papers

Abstract

We show that any set of n points in general position in the plane determines n(1-o(1)) pairwise crossing segments. The best previously known lower bound, Omega(root n), was proved more than 25 years ago by Aronov, Erdos, Goddard, Kreitman, Krugerman, Pach, and Schulman. Our proof is fully constructive, and extends to dense geometric graphs.

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https://infoscience.epfl.ch/entities/publication/1748937c-6797-4dc6-9dc9-8847b8ee9e42

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Related concepts (9)

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In graph theory, a planar graph is a graph that can be embedded in the plane, i.e., it can be drawn on the plane in such a way that its edges intersect only at their endpoints. In other words, it can be drawn in such a way that no edges cross each other. Such a drawing is called a plane graph or planar embedding of the graph. A plane graph can be defined as a planar graph with a mapping from every node to a point on a plane, and from every edge to a plane curve on that plane, such that the extreme points of each curve are the points mapped from its end nodes, and all curves are disjoint except on their extreme points.

Planar separator theorem

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