Classification of Planar Upward Embedding

  • Christopher Auer
  • Christian Bachmaier
  • Franz Josef Brandenburg
  • Andreas Gleißner
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7034)

Abstract

We consider planar upward drawings of directed graphs on arbitrary surfaces where the upward direction is defined by a vector field. This generalizes earlier approaches using surfaces with a fixed embedding in ℝ3 and introduces new classes of planar upward drawable graphs, where some of them even allow cycles. Our approach leads to a classification of planar upward embeddability.

In particular, we show the coincidence of the classes of planar upward drawable graphs on the sphere and on the standing cylinder. These classes coincide with the classes of planar upward drawable graphs with a homogeneous field on a cylinder and with a radial field in the plane.

A cyclic field in the plane introduces the new class RUP of upward drawable graphs, which can be embedded on a rolling cylinder. We establish strict inclusions for planar upward drawability on the plane, the sphere, the rolling cylinder, and the torus, even for acyclic graphs. Finally, upward drawability remains NP-hard for the standing cylinder and the torus; for the cylinder this was left as an open problem by Limaye et al.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Christopher Auer
    • 1
  • Christian Bachmaier
    • 1
  • Franz Josef Brandenburg
    • 1
  • Andreas Gleißner
    • 1
  1. 1.University of PassauPassauGermany

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