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Upward and Orthogonal Planarity are W[1]-Hard Parameterized by Treewidth

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Graph Drawing and Network Visualization (GD 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14466))

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Abstract

Upward planarity testing and Rectilinear planarity testing are central problems in graph drawing. It is known that they are both \(\textsf{NP}\)-complete, but \(\textsf{XP}\) when parameterized by treewidth. In this paper we show that these two problems are W[1]-hard parameterized by treewidth, which answers open problems posed in two earlier papers. The key step in our proof is an analysis of the All-or-Nothing Flow problem, a generalization of which was used as an intermediate step in the NP-completeness proof for both planarity testing problems. We prove that the flow problem is W[1]-hard parameterized by treewidth on planar graphs, and that the existing chain of reductions to the planarity testing problems can be adapted without blowing up the treewidth. Our reductions also show that the known \(n^{\mathcal {O}(\textsf{tw})}\)-time algorithms cannot be improved to run in time \(n^{o(\textsf{tw})}\) unless ETH fails.

Bart M. P. Jansen has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 803421, ReduceSearch).

Liana Khazaliya is supported by Vienna Science and Technology Fund (WWTF) [10.47379/ICT22029]; Austrian Science Fund (FWF) [Y1329]; European Union’s Horizon 2020 COFUND programme [LogiCS@TUWien, grant agreement No. 101034440].

G. Liotta and F. Montecchiani—This work was supported, in part, by MUR of Italy, under PRIN Project n. 2022ME9Z78 - NextGRAAL: Next-generation algorithms for constrained GRAph visuALization, and under PRIN Project n. 2022TS4Y3N - EXPAND: scalable algorithms for EXPloratory Analyses of heterogeneous and dynamic Networked Data.

Kirill Simonov acknowledges support by DFG Research Group ADYN via grant DFG 411362735.

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Notes

  1. 1.

    https://arxiv.org/abs/2309.01264.

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Acknowledgements

We acknowledge the fruitful working atmosphere of Dagstuhl Seminar 23162 “New Frontiers of Parameterized Complexity in Graph Drawing”, where this work was started.

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Authors and Affiliations

  1. Eindhoven University of Technology, Eindhoven, The Netherlands

    Bart M. P. Jansen

  2. Technische Universität Wien, Vienna, Austria

    Liana Khazaliya

  3. Universität Trier, Trier, Germany

    Philipp Kindermann

  4. University of Perugia, Perugia, Italy

    Giuseppe Liotta & Fabrizio Montecchiani

  5. Hasso Plattner Institute, University of Potsdam, Potsdam, Germany

    Kirill Simonov

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  1. Bart M. P. Jansen
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Correspondence to Liana Khazaliya .

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  1. University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Osnabrück University, Osnabrück, Germany

    Markus Chimani

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Jansen, B.M.P., Khazaliya, L., Kindermann, P., Liotta, G., Montecchiani, F., Simonov, K. (2023). Upward and Orthogonal Planarity are W[1]-Hard Parameterized by Treewidth. In: Bekos, M.A., Chimani, M. (eds) Graph Drawing and Network Visualization. GD 2023. Lecture Notes in Computer Science, vol 14466. Springer, Cham. https://doi.org/10.1007/978-3-031-49275-4_14

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  • DOI: https://doi.org/10.1007/978-3-031-49275-4_14

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