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Small Universal Point Sets for k-Outerplanar Graphs

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Abstract

A point set \(\mathcal{S} \subseteq \mathbb {R}^2\) is universal for a class \(\mathcal G\) of planar graphs if every graph of \(\mathcal{G}\) has a planar straight-line embedding on \(\mathcal{S}\). It is well-known that the integer grid is a quadratic-size universal point set for planar graphs, while the existence of a subquadratic universal point set still remains one of the most fascinating open problems in Graph Drawing. In this paper we make a major step towards a solution for this problem. Motivated by the fact that each point set of size n in general position is universal for the class of n-vertex outerplanar graphs, we concentrate our attention on k-outerplanar graphs. We prove that they admit an \(O(n \log n)\)-size universal point set in two distinct cases, namely when \(k=2\) (2-outerplanar graphs) and when k is unbounded but each outerplanarity level is restricted to be a simple cycle (simply-nested graphs).

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Notes

  1. The original definition in [10] is slightly different, as it allows the innermost level to be a tree and requires each internal face to be a 3-cycle.

  2. We thank the anonymous reviewer for pointing us to the prism drawings of outerplanar graphs.

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Authors and Affiliations

  1. Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, 72076, Tübingen, Germany

    Patrizio Angelini, Till Bruckdorfer & Michael Kaufmann

  2. Dipartimento di Ingegneria, Roma Tre University, 00146, Rome, Italy

    Giuseppe Di Battista, Vincenzo Roselli & Claudio Squarcella

  3. Institute of Theoretical Informatics, Karlsruhe Institute of Technology, 76131, Karlsruhe, Germany

    Tamara Mchedlidze

Authors
  1. Patrizio Angelini
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  2. Till Bruckdorfer
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  3. Giuseppe Di Battista
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  4. Michael Kaufmann
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  5. Tamara Mchedlidze
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  6. Vincenzo Roselli
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  7. Claudio Squarcella
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Correspondence to Patrizio Angelini.

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Editor in Charge: János Pach

This work has been supported by DFG Grant Ka812/17-1 and by MIUR Project “AMANDA” under PRIN 2012C4E3KT.

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Angelini, P., Bruckdorfer, T., Di Battista, G. et al. Small Universal Point Sets for k-Outerplanar Graphs. Discrete Comput Geom 60, 430–470 (2018). https://doi.org/10.1007/s00454-018-0009-x

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  • DOI: https://doi.org/10.1007/s00454-018-0009-x

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