Geometric Biplane Graphs II: Graph Augmentation

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.