The density of iterated crossing points and a gap result for triangulations of finite point sets

TL;DR

This paper investigates the geometric properties of point sets and their triangulations, establishing a gap theorem that links the density of crossing points to the dilation of plane graphs, with implications for graph embedding quality.

Contribution

It introduces a gap theorem connecting the density of iterated crossing points in point sets to the dilation thresholds of plane graphs, providing new bounds and constructions.

Findings

Infinite crossing point sets imply a dilation gap greater than 1.

Constructed point set Q requires dilation > 1.0000047.

Finite graphs cannot contain certain dense crossing point sets with low dilation.

Abstract

Consider a plane graph G, drawn with straight lines. For every pair a,b of vertices of G, we compare the shortest-path distance between a and b in G (with Euclidean edge lengths) to their actual distance in the plane. The worst-case ratio of these two values, for all pairs of points, is called the dilation of G. All finite plane graphs of dilation 1 have been classified. They are closely related to the following iterative procedure. For a given point set P in R^2, we connect every pair of points in P by a line segment and then add to P all those points where two such line segments cross. Repeating this process infinitely often, yields a limit point set P*. The main result of this paper is the following gap theorem: For any finite point set P in the plane for which the above P* is infinite, there exists a threshold t > 1 such that P is not contained in the vertex set of any finite plane graph of dilation at most t. We construct a concrete point set Q such that any planar graph that contains this set amongst its vertices must have a dilation larger than 1.0000047.