# Geometric Biplane Graphs II: Graph Augmentation

## TL;DR

This paper investigates the connectivity properties of biplane graphs on point sets, demonstrating the existence of highly connected biplane graphs and limitations on augmenting plane graphs to higher connectivity levels.

## Contribution

It establishes new bounds on the connectivity of biplane graphs and shows how certain plane graphs can or cannot be augmented to achieve higher connectivity.

## Key findings

- Every large point set admits a 5-connected biplane graph.
- Some large point sets do not admit a 6-connected biplane graph.
- Most plane graphs can be augmented into a 4-connected biplane graph.

## Abstract

We study biplane graphs drawn on a finite point set $S$ in the plane in general position. This is the family of geometric graphs whose vertex set is $S$ and which can be decomposed into two plane graphs. We show that every sufficiently large point set admits a 5-connected biplane graph and that there are arbitrarily large point sets that do not admit any 6-connected biplane graph. Furthermore, we show that every plane graph (other than a wheel or a fan) can be augmented into a 4-connected biplane graph. However, there are arbitrarily large plane graphs that cannot be augmented to a 5-connected biplane graph by adding pairwise noncrossing edges.