Geometric Biplane Graphs I: Maximal Graphs

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 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 biplane graphs such as the maximum number of edges and the largest minimum degree of biplane graphs over n-element point sets. In a companion paper we study how to draw a biplane graph on a given point set S, or how to augment a given biplane graph on S, in such a way that the resulting graph is biplane and has good connectivity properties.