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Universal Slope Sets for 1-Bend Planar Drawings

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Abstract

We prove that every set of \(\varDelta -1\) slopes is 1-bend universal for the planar graphs with maximum vertex degree \(\varDelta \). This means that any planar graph with maximum degree \(\varDelta \) admits a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges can be chosen in any given set of \(\varDelta -1\) slopes. Our result improves over previous literature in three ways: Firstly, it improves the known upper bound of \(\frac{3}{2} (\varDelta -1)\) on the 1-bend planar slope number; secondly, the previously known algorithms compute 1-bend planar drawings by using sets of \(O(\varDelta )\) slopes that may vary depending on the input graph; thirdly, while these algorithms typically minimize the slopes at the expenses of constructing drawings with poor angular resolution, we can compute drawings whose angular resolution is at least \(\frac{\pi }{\varDelta -1}\), which is worst-case optimal up to a factor of \(\frac{3}{4}\). Our proofs are constructive and give rise to a linear-time drawing algorithm.

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Notes

  1. Note that, formally, for a straight line \(\ell \), the angle that a horizontal line needs to be rotated counter-clockwise in order to make it overlap with \(\ell \) is the angle of incline of \(\ell \), while its slopes is the tangent of this angle. However, for simplicity reasons, and similarly as in previous papers (see, e.g., [16, 19, 20]), we refer directly to the angle of incline of \(\ell \) as its slope.

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Acknowledgements

This work started at the 19th Korean Workshop on Computational Geometry. We wish to thank the organizers and the participants for creating a pleasant and stimulating atmosphere and in particular Fabian Lipp and Boris Klemz for useful discussions. We also gratefully thank the anonymous reviewers for their useful comments.

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Authors and Affiliations

  1. Wilhelm-Schickhard-Institut für Informatik, Universität Tübingen, Tübingen, Germany

    Patrizio Angelini & Michael A. Bekos

  2. Dipartimento di Ingegneria, University of Perugia, Perugia, Italy

    Giuseppe Liotta & Fabrizio Montecchiani

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  1. Patrizio Angelini
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  2. Michael A. Bekos
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Correspondence to Fabrizio Montecchiani.

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An extended abstract of this paper has been presented at the 33rd International Symposium on Computational Geometry (SoCG 2017). Research supported in part by the project: “Algoritmi e sistemi di analisi visuale di reti complesse e di grandi dimensioni” - Ricerca di Base 2018, Dipartimento di Ingegneria dell’Università degli Studi di Perugia, and by the DFG Grants Ka812/17-1 and Ka812/18-1.

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Angelini, P., Bekos, M.A., Liotta, G. et al. Universal Slope Sets for 1-Bend Planar Drawings. Algorithmica 81, 2527–2556 (2019). https://doi.org/10.1007/s00453-018-00542-9

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