On the Density of Maximal 1-Planar Graphs

  • Franz J. Brandenburg
  • David Eppstein
  • Andreas Gleißner
  • Michael T. Goodrich
  • Kathrin Hanauer
  • Josef Reislhuber
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

Abstract

A graph is 1-planar if it can be drawn in the plane such that each edge is crossed at most once. It is maximal 1-planar if the addition of any edge violates 1-planarity.

Maximal 1-planar graphs have at most 4n − 8 edges. We show that there are sparse maximal 1-planar graphs with only \(\frac{45}{17} n + \mathcal{O}(1)\) edges. With a fixed rotation system there are maximal 1-planar graphs with only \(\frac{7}{3} n + \mathcal{O}(1)\) edges. This is sparser than maximal planar graphs. There cannot be maximal 1-planar graphs with less than \(\frac{21}{10} n - \mathcal{O}(1)\) edges and less than \(\frac{28}{13} n - \mathcal{O}(1)\) edges with a fixed rotation system. Furthermore, we prove that a maximal 1-planar rotation system of a graph uniquely determines its 1-planar embedding.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Franz J. Brandenburg
    • 1
  • David Eppstein
    • 2
  • Andreas Gleißner
    • 1
  • Michael T. Goodrich
    • 2
  • Kathrin Hanauer
    • 1
  • Josef Reislhuber
    • 1
  1. 1.University of PassauPassauGermany
  2. 2.Department of Computer ScienceUniversity of CaliforniaIrvineUSA

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