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arXiv · 1702.01275

Geometric Biplane Graphs I: Maximal Graphs

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 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.

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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 I: Maximal Graphs. https://doi.org/10.1007/s00373-015-1546-1

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