A Tipping Point for the Planarity of Small and Medium Sized Graphs

TL;DR

This study empirically investigates how the density of small to medium-sized random graphs influences their planarity, revealing a sharp transition similar to the known asymptotic behavior in large graphs.

Contribution

It demonstrates that a sharp planarity transition occurs in small and medium graphs, extending the understanding of graph planarity beyond asymptotic cases.

Findings

Sharp transition in planarity for small/medium graphs at certain densities

Similar transition observed for outerplanarity and near-planarity

Empirical evidence supports the existence of a 'tipping point' in graph properties

Abstract

This paper presents an empirical study of the relationship between the density of small-medium sized random graphs and their planarity. It is well known that, when the number of vertices tends to infinite, there is a sharp transition between planarity and non-planarity for edge density d=0.5. However, this asymptotic property does not clarify what happens for graphs of reduced size. We show that an unexpectedly sharp transition is also exhibited by small and medium sized graphs. Also, we show that the same "tipping point" behavior can be observed for some restrictions or relaxations of planarity (we considered outerplanarity and near-planarity, respectively).