Graphs and Combinatorics

, Volume 31, Issue 2, pp 427–452 | Cite as

Geometric Biplane Graphs II: Graph Augmentation

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

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.

Keywords

Geometric graphs Biplane graphs \(k\)-connected graphs Graph augmentation 

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Copyright information

© Springer Japan 2015

Authors and Affiliations

  • Alfredo García
    • 1
  • Ferran Hurtado
    • 2
  • Matias Korman
    • 3
  • Inês Matos
    • 4
  • Maria Saumell
    • 5
  • Rodrigo I. Silveira
    • 2
  • Javier Tejel
    • 1
  • Csaba D. Tóth
    • 6
  1. 1.Departamento de Métodos Estadísticos, IUMAUniversidad de ZaragozaZaragozaSpain
  2. 2.Departament de Matemàtica Aplicada IIUPCBarcelonaSpain
  3. 3.Erato Kawarabayashi Large Graph Project, JSTNational Institute of Informatics (NII)TokyoJapan
  4. 4.Departamento de Matemática and CIDMAUniversidade de AveiroAveiroPortugal
  5. 5.Department of Mathematics and European Centre of Excellence NTIS (New Technologies for the Information Society)University of West BohemiaPilsenCzech Republic
  6. 6.Department of MathematicsCalifornia State University NorthridgeLos AngelesUSA

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