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The Price of Upwardness

Authors Patrizio Angelini , Therese Biedl , Markus Chimani , Sabine Cornelsen , Giordano Da Lozzo , Seok-Hee Hong , Giuseppe Liotta , Maurizio Patrignani , Sergey Pupyrev , Ignaz Rutter , Alexander Wolff

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Author Details

Patrizio Angelini
  • John Cabot University, Rome, Italy
Therese Biedl
  • University of Waterloo, Canada
Markus Chimani
  • University of Osnabrück, Germany
Sabine Cornelsen
  • University of Konstanz, Germany
Giordano Da Lozzo
  • Roma Tre University, Italy
Seok-Hee Hong
  • University of Sydney, Australia
Giuseppe Liotta
  • University of Perugia, Italy
Maurizio Patrignani
  • Roma Tre University, Italy
Sergey Pupyrev
  • Menlo Park, CA, USA
Ignaz Rutter
  • University of Passau, Germany
Alexander Wolff
  • University of Würzburg, Germany

Funding

Research by {Da Lozzo}, {Liotta}, and {Patrignani} was supported, in part, by MUR of Italy (PRIN Project no. 2022ME9Z78 - NextGRAAL and PRIN Project no. 2022TS4Y3N - EXPAND). Research by {Liotta} was supported in part by MUR PON Proj. ARS01_00540. Research by Rutter was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 541433306. Research by Biedl was supported by NSERC FRN RGPIN-2020-03958.

Acknowledgements

This work was initiated at the Dagstuhl Seminar "Beyond-Planar Graphs: Models, Structures and Geometric Representations" (No. 24062), February 2024.

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Patrizio Angelini, Therese Biedl, Markus Chimani, Sabine Cornelsen, Giordano Da Lozzo, Seok-Hee Hong, Giuseppe Liotta, Maurizio Patrignani, Sergey Pupyrev, Ignaz Rutter, and Alexander Wolff. The Price of Upwardness. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.13

Abstract

Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward k-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most k times for some integer k ≥ 1. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that upward-k-planarity testing is NP-complete already for k = 1 and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward 1-planar drawing in which all vertices are incident to the outer face.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Graph theory
Keywords
  • upward drawings
  • beyond planarity
  • upward k-planarity
  • upward outer-1-planarity

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