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On k-Plane Insertion into Plane Drawings

Authors Julia Katheder , Philipp Kindermann , Fabian Klute , Irene Parada , Ignaz Rutter

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Author Details

Julia Katheder
  • Universität Tübingen, Germany
Philipp Kindermann
  • Universität Trier, Germany
Fabian Klute
  • Universitat Politècnica de Catalunya, Barcelona, Spain
Irene Parada
  • Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
Ignaz Rutter
  • Universität Passau, Germany

Funding

  • Katheder, Julia: Funded by the Deutsche Forschungsgemeinschaft (DFG) - 364468267.
  • Klute, Fabian: F. K. is supported by a “María Zambrano grant for attracting international talent” and by grant PID2019-104129GB-I00 funded by MICIU/AEI/10.13039/501100011033.
  • Parada, Irene: I. P. is a Serra Húnter Fellow. Partially supported by grant 2021UPC-MS-67392 funded by the Spanish Ministry of Universities and the European Union (NextGenerationEU) and by grant PID2019-104129GB-I00 funded by MICIU/AEI/10.13039/501100011033.
  • Rutter, Ignaz: Funded by the Deutsche Forschungsgemeinschaft (DFG) - 541433306.

Cite AsGet BibTex

Julia Katheder, Philipp Kindermann, Fabian Klute, Irene Parada, and Ignaz Rutter. On k-Plane Insertion into Plane Drawings. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 35:1-35:11, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.35

Abstract

We introduce the k-Plane Insertion into Plane drawing (k-PIP) problem: given a plane drawing of a planar graph G and a set F of edges, insert the edges in F into the drawing such that the resulting drawing is k-plane. In this paper, we show that the problem is NP-complete for every k ≥ 1, even when G is biconnected and the set F of edges forms a matching or a path. On the positive side, we present a linear-time algorithm for the case that k = 1 and G is a triangulation.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatoric problems
Keywords
  • Graph drawing
  • edge insertion
  • k-planarity

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References

  1. Patrizio Angelini, Giuseppe Di Battista, Fabrizio Frati, Vít Jelínek, Jan Kratochvíl, Maurizio Patrignani, and Ignaz Rutter. Testing planarity of partially embedded graphs. ACM Transactions on Algorithms, 11(4):32:1endash32:42, 2015. URL: https://doi.org/10.1145/2629341.
  2. Patrizio Angelini, Ignaz Rutter, and Sandhya T. P. Extending partial orthogonal drawings. In Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD'20), volume 12590 of LNCS, page 265endash278. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-68766-3_21.
  3. Alan Arroyo, Martin Derka, and Irene Parada. Extending simple drawings. In Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD)'19, volume 11904 of LNCS, pages 230-243. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-35802-0_18.
  4. Alan Arroyo, Fabian Klute, Irene Parada, Birgit Vogtenhuber, Raimund Seidel, and Tilo Wiedera. Inserting one edge into a simple drawing is hard. Discrete & Computational Geometry, 69(3):745endash770, 2023. URL: https://doi.org/10.1007/S00454-022-00394-9.
  5. Guido Brückner and Ignaz Rutter. Partial and constrained level planarity. In Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms (SODA'17), page 2000endash2011. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.130.
  6. Erin W. Chambers, David Eppstein, Michael T. Goodrich, and Maarten Löffler. Drawing graphs in the plane with a prescribed outer face and polynomial area. Journal of Graph Algorithms and Applications, 16(2):243endash259, 2012. URL: https://doi.org/10.7155/jgaa.00257.
  7. Timothy M. Chan, Fabrizio Frati, Carsten Gutwenger, Anna Lubiw, Petra Mutzel, and Marcus Schaefer. Drawing partially embedded and simultaneously planar graphs. Journal of Graph Algorithms and Applications, 19(2):681endash706, 2015. URL: https://doi.org/10.7155/jgaa.00375.
  8. Markus Chimani, Carsten Gutwenger, Petra Mutzel, and Christian Wolf. Inserting a vertex into a planar graph. In Proceedings of the 20th Annual ACM-SIAM Symposium on Discrete Algorithms, (SODA'09), page 375endash383. SIAM, 2009. URL: https://doi.org/10.1137/1.9781611973068.42.
  9. Markus Chimani and Petr Hlinený. Inserting multiple edges into a planar graph. Journal of Graph Algorithms and Applications, 27(6):489endash522, 2023. URL: https://doi.org/10.7155/JGAA.00631.
  10. Mark de Berg and Amirali Khosravi. Optimal binary space partitions for segments in the plane. International Journal of Computational Geometry and Applications, 22(3):187endash206, 2012. URL: https://doi.org/10.1142/S0218195912500045.
  11. Walter Didimo, Giuseppe Liotta, and Fabrizio Montecchiani. A survey on graph drawing beyond planarity. ACM Computing Surveys, 52(1):4:1endash4:37, 2019. URL: https://doi.org/10.1145/3301281.
  12. Reinhard Diestel. Graph Theory, 5th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. URL: https://doi.org/10.1007/978-3-662-53622-3.
  13. Eduard Eiben, Robert Ganian, Thekla Hamm, Fabian Klute, and Martin Nöllenburg. Extending nearly complete 1-planar drawings in polynomial time. In Proceedings of the 45th International Symposium on Mathematical Foundations of Computer Science, (MFCS'20), volume 170 of LIPIcs, page 31:1endash31:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.31.
  14. Eduard Eiben, Robert Ganian, Thekla Hamm, Fabian Klute, and Martin Nöllenburg. Extending partial 1-planar drawings. In Proceedings of the 47th International Colloquium on Automata, Languages, and Programming (ICALP'20), volume 168 of LIPIcs, page 43:1endash43:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.43.
  15. Robert Ganian, Thekla Hamm, Fabian Klute, Irene Parada, and Birgit Vogtenhuber. Crossing-optimal extension of simple drawings. In 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference), volume 198 of LIPIcs, page 72:1endash72:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPICS.ICALP.2021.72.
  16. Carsten Gutwenger, Petra Mutzel, and René Weiskircher. Inserting an edge into a planar graph. Algorithmica, 41(4):289endash308, 2005. URL: https://doi.org/10.1007/S00453-004-1128-8.
  17. Thekla Hamm and Petr Hliněný. Parameterised partially-predrawn crossing number. In Proceedings of the 38th International Symposium on Computational Geometry (SoCG'22), volume 224 of LIPIcs, page 46:1endash46:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.46.
  18. Seok-Hee Hong and Takeshi Tokuyama, editors. Beyond Planar Graphs, Communications of NII Shonan Meetings. Springer, 2020. URL: https://doi.org/10.1007/978-981-15-6533-5.
  19. Julia Katheder, Philipp Kindermann, Fabian Klute, Irene Parada, and Ignaz Rutter. On k-plane insertion into plane drawings. CoRR, abs/2402.14552, 2024. URL: https://doi.org/10.48550/arXiv.2402.14552.
  20. Donald E. Knuth and Arvind Raghunathan. The problem of compatible representatives. SIAM Journal on Discrete Mathematics, 5(3):422endash427, 1992. URL: https://doi.org/10.1137/0405033.
  21. Giordano Da Lozzo, Giuseppe Di Battista, and Fabrizio Frati. Extending upward planar graph drawings. Computational Geometry: Theory and Applications, 91:101668, 2020. URL: https://doi.org/10.1016/j.comgeo.2020.101668.
  22. Tamara Mchedlidze, Martin Nöllenburg, and Ignaz Rutter. Extending convex partial drawings of graphs. Algorithmica, 76(1):47endash67, 2016. URL: https://doi.org/10.1007/s00453-015-0018-6.
  23. Petra Mutzel and Thomas Ziegler. The constrained crossing minimization problem. In Jan Kratochvíl, editor, Proceedings of the 7th International Symposium on Graph Drawing (GD'99), volume 1731 of Lecture Notes in Computer Science, page 175endash185. Springer, 1999. URL: https://doi.org/10.1007/3-540-46648-7_18.
  24. Petra Mutzel and Thomas Ziegler. The constrained crossing minimization problem a first approach. In Proceedings of the 1999 International Conference on Operations Research, page 125endash134. Springer, 1999. URL: https://doi.org/10.1007/978-3-642-58409-1_11.
  25. Maurizio Patrignani. On extending a partial straight-line drawing. International Journal of Foundations of Computer Science, 17(5):1061endash1070, 2006. URL: https://doi.org/10.1142/S0129054106004261.
  26. Thomas Ziegler. Crossing minimization in automatic graph drawing. PhD thesis, Saarland University, Saarbrücken, Germany, 2001. URL: https://d-nb.info/961610808.
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