An Omega(n^2) Lower Bound for Random Universal Sets for Planar Graphs

TL;DR

This paper proves that any random point set in the plane that can embed all n-vertex planar graphs with high probability must have at least on the order of n^2 points, establishing a fundamental lower bound.

Contribution

It establishes an Omega(n^2) lower bound on the size of random universal point sets for planar graphs, showing limitations of probabilistic methods for this problem.

Findings

Random universal sets require at least Omega(n^2) points.

Probabilistic methods cannot achieve o(n^2) bounds for universal sets.

High probability embeddings demand large point sets.

Abstract

A set $U\subseteq \reals^2$ is $n$-universal if all $n$-vertex planar graphs have a planar straight-line embedding into $U$. We prove that if $Q \subseteq \reals^2$ consists of points chosen randomly and uniformly from the unit square then $Q$ must have cardinality $\Omega(n^2)$ in order to be $n$-universal with high probability. This shows that the probabilistic method, at least in its basic form, cannot be used to establish an $o(n^2)$ upper bound on universal sets.