Geometric Biplane Graphs II: Graph Augmentation

• Published in: Graphs and combinatorics Vol. 31; no. 2; pp. 427 - 452
• Main Authors: García, Alfredo, Hurtado, Ferran, Korman, Matias, Matos, Inês, Saumell, Maria, Silveira, Rodrigo I., Tejel, Javier, Tóth, Csaba D.
• Format: Journal Article Publication
• Language: English
• Published: Tokyo Springer Japan 01.03.2015; Springer Nature B.V
• Subjects: 30 Functions of a complex variable; 30C Geometric function theory; Augmentation; Biplane graphs; Biplanes; Classificació AMS; Combinatorial analysis; Combinatorics; COMMON ANCESTORS; CONNECTIVITY; Engineering Design; Geometria; Geometria computacional; Geometric graphs; Geometric group theory; Geometry; Graph augmentation; Graph theory; Graphs; k-connected graphs; Matemàtiques i estadística; Mathematical analysis; Mathematics; Mathematics and Statistics; Original Paper; Planes; Texts; Wheels; Àrees temàtiques de la UPC; Biplane graphs; connected graphs; Geometric graphs; Graph augmentation
• ISSN: 0911-0119; 1435-5914
• DOI: 10.1007/s00373-015-1547-0

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.