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.

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
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    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
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    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
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    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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