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Planar Lombardi Drawings of Outerpaths

  • Maarten Löffler
  • Martin Nöllenburg
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7704)

Introduction

A Lombardi drawing of a graph is a drawingwhere edges are represented by circular arcs that meet at each vertex v with perfect angular resolution 360°/deg(v) [3]. It is known that Lombardi drawings do not always exist, and in particular, that planar Lombardi drawings of planar graphs do not always exist [1], even when the embedding is not fixed. Existence of planar Lombardi drawings is known for restricted classes of graphs, such as subcubic planar graphs [4], trees [2], Halin graphs and some very symmetric planar graphs [3]. On the other hand, all 2-degenerate graphs, including all outerplanar graphs, have Lombardi drawings, but not necessarily planar ones [3]. One question that was left open is whether outerplanar graphs always have planar Lombardi drawings or not.

In this note, we report that the answer is “yes” for a more restricted subclass: the outerpaths, i.e., outerplanar graphs whose weak dual is a path. We sketch an algorithm that produces an outerplanar Lombardi drawing of any outerpath, in linear time.

Keywords

Planar Graph Outerplanar Graph Edge Crossing Halin Graph Subcubic Graph 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

References

  1. 1.
    Duncan, C.A., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Löffler, M.: Planar and Poly-arc Lombardi Drawings. In: van Kreveld, M., Speckmann, B. (eds.) GD 2011. LNCS, vol. 7034, pp. 308–319. Springer, Heidelberg (2012)Google Scholar
  2. 2.
    Duncan, C.A., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Nöllenburg, M.: Drawing Trees with Perfect Angular Resolution and Polynomial Area. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 183–194. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  3. 3.
    Duncan, C.A., Eppstein, D., Goodrich, M.T., Kobourov, S.G., Nöllenburg, M.: Lombardi drawings of graphs. J. Graph Algorithms and Applications 16(1), 85–108 (2012)MathSciNetMATHCrossRefGoogle Scholar
  4. 4.
    Eppstein, D.: Planar Lombardi Drawings for Subcubic Graphs. In: Didimo, W., Patrignani, M. (eds.) GD 2012. LNCS, vol. 7704, pp. 126–137. Springer, Heidelberg (2013) To appear arXiv:1206.6142 Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Maarten Löffler
    • 1
  • Martin Nöllenburg
    • 2
  1. 1.Dept. of Information and Computing SciencesUtrecht UniversityThe Netherlands
  2. 2.Institut für Theoretische InformatikKITGermany

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