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A Universal Slope Set for 1-Bend Planar Drawings

Authors Patrizio Angelini, Michael A. Bekos, Giuseppe Liotta, Fabrizio Montecchiani

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Patrizio Angelini
Michael A. Bekos
Giuseppe Liotta
Fabrizio Montecchiani

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Patrizio Angelini, Michael A. Bekos, Giuseppe Liotta, and Fabrizio Montecchiani. A Universal Slope Set for 1-Bend Planar Drawings. In 33rd International Symposium on Computational Geometry (SoCG 2017). Leibniz International Proceedings in Informatics (LIPIcs), Volume 77, pp. 9:1-9:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2017)
https://doi.org/10.4230/LIPIcs.SoCG.2017.9

Abstract

We describe a set of Delta-1 slopes that are universal for 1-bend planar drawings of planar graphs of maximum degree Delta>=4; this establishes a new upper bound of Delta-1 on the 1-bend planar slope number. By universal we mean that every planar graph of degree Delta has a planar drawing with at most one bend per edge and such that the slopes of the segments forming the edges belong to the given set of slopes. This improves over previous results in two ways: Firstly, the best previously known upper bound for the 1-bend planar slope number was 3/2(Delta-1) (the known lower bound being 3/4(Delta-1)); secondly, all the known algorithms to construct 1-bend planar drawings with O(Delta) slopes use a different set of slopes for each graph and can have bad angular resolution, while our algorithm uses a universal set of slopes, which also guarantees that the minimum angle between any two edges incident to a vertex is pi/(Delta-1).
Keywords
  • Slope number
  • 1-bend drawings
  • planar graphs
  • angular resolution

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