Plane and Planarity Thresholds for Random Geometric Graphs

TL;DR

This paper investigates the thresholds for various geometric and topological properties of random geometric graphs, including planarity, connectivity, and subgraph configurations, based on the Euclidean distance parameter.

Contribution

It establishes new threshold functions for planarity, non-crossing edges, and specific subgraph structures in random geometric graphs.

Findings

Threshold for connectivity on k points: n^{-k/(2k-2)}

Threshold for planarity: n^{-2/3}

Threshold for being planar: n^{-5/8}

Abstract

A random geometric graph, $G(n,r)$, is formed by choosing $n$ points independently and uniformly at random in a unit square; two points are connected by a straight-line edge if they are at Euclidean distance at most $r$. For a given constant $k$, we show that $n^{\frac{-k}{2k-2}}$ is a distance threshold function for $G(n,r)$ to have a connected subgraph on $k$ points. Based on this, we show that $n^{-2/3}$ is a distance threshold for $G(n,r)$ to be plane, and $n^{-5/8}$ is a distance threshold to be planar. We also investigate distance thresholds for $G(n,r)$ to have a non-crossing edge, a clique of a given size, and an independent set of a given size.