Existence of a plane without edge crossings in projections of the random geometric graph

TL;DR

This paper proves that for large random geometric graphs with small connection radius, the probability of finding a plane projection without edge crossings approaches one, and analyzes the likelihood of such planes among random samples.

Contribution

It establishes the asymptotic certainty of a crossing-free projection in large graphs with small connection radius and quantifies the probability among random plane choices.

Findings

Probability of a crossing-free plane tends to one as vertex density increases.

Asymptotic probability of finding such a plane among random samples is derived.

Results depend on the connection radius being below a certain threshold.

Abstract

Consider a random geometric graph $G$ with a vertex set defined by a Poisson point process with intensity $t>0$ in a convex body. We can generate a drawing of the graph by projecting the construction onto some plane $L$. Choosing different planes leads to different drawings, and in particular, potentially more or fewer edge crossings. In this paper, we prove that if the connection radius is smaller than a given threshold, the probability that there exists a plane with zero crossings tends to one as $t\to \infty$. We also state the asymptotic probability that such a plane is found after considering a given number of randomly chosen planes.