# Geometric Biplane Graphs I: Maximal Graphs

## TL;DR

This paper investigates the properties of biplane graphs on finite point sets, focusing on maximal graphs, their edge counts, and connectivity, and provides algorithms for their augmentation.

## Contribution

It introduces the concept that maximal biplane graphs may differ in edges and offers an efficient algorithm to make a biplane graph maximal.

## Key findings

- Maximal biplane graphs can differ in the number of edges.
- An efficient algorithm exists for augmenting biplane graphs to maximality.
- Extremal properties like maximum edges and connectivity are characterized.

## Abstract

We study biplane graphs drawn on a finite planar point set $S$ in general position. This is the family of geometric graphs whose vertex set is $S$ and can be decomposed into two plane graphs. We show that two maximal biplane graphs---in the sense that no edge can be added while staying biplane---may differ in the number of edges, and we provide an efficient algorithm for adding edges to a biplane graph to make it maximal. We also study extremal properties of maximal biplane graphs such as the maximum number of edges and the largest maximum connectivity over $n$-element point sets.