, Volume 81, Issue 6, pp 2527–2556 | Cite as

Universal Slope Sets for 1-Bend Planar Drawings

  • Patrizio Angelini
  • Michael A. Bekos
  • Giuseppe Liotta
  • Fabrizio MontecchianiEmail author


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.


Graph drawing Slope number 1-Bend planar drawings 



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. 1.Wilhelm-Schickhard-Institut für InformatikUniversität Tübingen TübingenGermany
  2. 2.Dipartimento di IngegneriaUniversity of Perugia PerugiaItaly

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