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arXiv · 1702.01277

Geometric Biplane Graphs II: Graph Augmentation

Alfredo García·Ferran Hurtado·Matias Korman·Inês Matos·Maria Saumell·Rodrigo I. Silveira·Javier Tejel·Csaba D. Tóth
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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.

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Alfredo García, Ferran Hurtado, Matias Korman, Inês Matos, Maria Saumell, Rodrigo I. Silveira, Javier Tejel, Csaba D. Tóth. 2017-02-04. Geometric Biplane Graphs II: Graph Augmentation. https://doi.org/10.1007/s00373-015-1547-0

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