Graphs and Combinatorics

, Volume 31, Issue 2, pp 427–452

# Geometric Biplane Graphs II: Graph Augmentation

• Alfredo García
• Matias Korman
• Inês Matos
• Maria Saumell
• Rodrigo I. Silveira
• Javier Tejel
• Csaba D. Tóth
Original Paper

## 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.

### Keywords

Geometric graphs Biplane graphs $$k$$-connected graphs Graph augmentation

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## Authors and Affiliations

• Alfredo García
• 1
• 2
• Matias Korman
• 3
• Inês Matos
• 4
• Maria Saumell
• 5
• Rodrigo I. Silveira
• 2
• Javier Tejel
• 1
• Csaba D. Tóth
• 6