International Symposium on Graph Drawing

GD 2008: Graph Drawing pp 302-312

Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing

  • Vladimir P. Korzhik
  • Bojan Mohar
Conference paper

DOI: 10.1007/978-3-642-00219-9_29

Volume 5417 of the book series Lecture Notes in Computer Science (LNCS)
Cite this paper as:
Korzhik V.P., Mohar B. (2009) Minimal Obstructions for 1-Immersions and Hardness of 1-Planarity Testing. In: Tollis I.G., Patrignani M. (eds) Graph Drawing. GD 2008. Lecture Notes in Computer Science, vol 5417. Springer, Berlin, Heidelberg

Abstract

A graph is 1-planar if it can be drawn on the plane so that each edge is crossed by no more than one other edge. A non-1-planar graph G is minimal if the graph G − e is 1-planar for every edge e of G. We construct two infinite families of minimal non-1-planar graphs and show that for every integer n ≥ 63, there are at least \(2^{\frac{n}{4}-\frac{54}{4}}\) non-isomorphic minimal non-1-planar graphs of order n. It is also proved that testing 1-planarity is NP-complete. As an interesting consequence we obtain a new, geometric proof of NP-completeness of the crossing number problem, even when restricted to cubic graphs. This resolves a question of Hliněný.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Vladimir P. Korzhik
    • 1
  • Bojan Mohar
    • 2
  1. 1.National Academy of ScienceNational University of Chernivtsi and Institute of APMMLvivUkraine
  2. 2.Department of MathematicsSimon Fraser UniversityBurnabyCanada