Document Open Access Logo

Parameterised Partially-Predrawn Crossing Number

Authors Thekla Hamm , Petr Hliněný

PDF
Thumbnail PDF

File

LIPIcs.SoCG.2022.46.pdf
  • Filesize: 1.17 MB
  • 15 pages

Document Identifiers

Author Details

Thekla Hamm
  • Algorithms and Complexity Group, TU Wien, Austria
Petr Hliněný
  • Faculty of Informatics, Masaryk University, Brno, Czech Republic

Funding

  • Hamm, Thekla: Supported by the Austrian Science Fund (projects P31336, Y1329, and W1255-N23).
  • Hliněný, Petr: Supported by the {Czech Science Foundation}, project no. {20-04567S}.

Cite AsGet BibTex

Thekla Hamm and Petr Hliněný. Parameterised Partially-Predrawn Crossing Number. In 38th International Symposium on Computational Geometry (SoCG 2022). Leibniz International Proceedings in Informatics (LIPIcs), Volume 224, pp. 46:1-46:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2022)
https://doi.org/10.4230/LIPIcs.SoCG.2022.46

Abstract

Inspired by the increasingly popular research on extending partial graph drawings, we propose a new perspective on the traditional and arguably most important geometric graph parameter, the crossing number. Specifically, we define the partially predrawn crossing number to be the smallest number of crossings in any drawing of a graph, part of which is prescribed on the input (not counting the prescribed crossings). Our main result - an FPT-algorithm to compute the partially predrawn crossing number - combines advanced ideas from research on the classical crossing number and so called partial planarity in a very natural but intricate way. Not only do our techniques generalise the known FPT-algorithm by Grohe for computing the standard crossing number, they also allow us to substantially improve a number of recent parameterised results for various drawing extension problems.

Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
  • Theory of computation → Computational geometry
Keywords
  • Crossing Number
  • Drawing Extension
  • Partial Planarity
  • Parameterised Complexity

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Related Versions

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 Trans. Algorithms, 11(4), April 2015. URL: https://doi.org/10.1145/2629341.
  2. Stefan Arnborg, Jens Lagergren, and Detlef Seese. Easy problems for tree-decomposable graphs. J. Algorithms, 12(2):308-340, 1991. URL: https://doi.org/10.1016/0196-6774(91)90006-K.
  3. Sergio Cabello. Hardness of approximation for crossing number. Discrete Comput. Geom., 49(2):348-358, March 2013. Google Scholar
  4. Sergio Cabello and Bojan Mohar. Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM J. Comput., 42(5):1803-1829, January 2013. Google Scholar
  5. Katrin Casel, Henning Fernau, Mehdi Khosravian Ghadikolaei, Jérôme Monnot, and Florian Sikora. On the complexity of solution extension of optimization problems. Theoretical Computer Science, 2021. URL: https://doi.org/10.1016/j.tcs.2021.10.017.
  6. Markus Chimani and Petr Hliněný. Inserting multiple edges into a planar graph. In SoCG, volume 51 of LIPIcs, pages 30:1-30:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2016. Google Scholar
  7. Bruno Courcelle. The monadic second-order logic of graphs. I. recognizable sets of finite graphs. Inf. Comput., 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  8. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michał Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  9. Reinhard Diestel. Graph Theory, 4th Edition, volume 173 of Graduate texts in mathematics. Springer, 2012. Google Scholar
  10. Rodney G. Downey and Michael R. Fellows. Fundamentals of Parameterized Complexity. Texts in Computer Science. Springer, 2013. URL: https://doi.org/10.1007/978-1-4471-5559-1.
  11. Zdenek Dvořák, Petr Hliněný, and Bojan Mohar. Structure and generation of crossing-critical graphs. In SoCG, volume 99 of LIPIcs, pages 33:1-33:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2018. Google Scholar
  12. Eduard Eiben, Robert Ganian, Thekla Hamm, Fabian Klute, and Martin Nöllenburg. Extending nearly complete 1-planar drawings in polynomial time. In Javier Esparza and Daniel Král', editors, MFCS 2020, volume 170 of LIPIcs, pages 31:1-31:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.31.
  13. Eduard Eiben, Robert Ganian, Thekla Hamm, Fabian Klute, and Martin Nöllenburg. Extending partial 1-planar drawings. In Artur Czumaj, Anuj Dawar, and Emanuela Merelli, editors, ICALP 2020, volume 168 of LIPIcs, pages 43:1-43:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ICALP.2020.43.
  14. Eduard Eiben, Robert Ganian, Thekla Hamm, and O-joung Kwon. Measuring what matters: A hybrid approach to dynamic programming with treewidth. Journal of Computer and System Sciences, 121:57-75, 2021. URL: https://doi.org/10.1016/j.jcss.2021.04.005.
  15. Robert Ganian, Thekla Hamm, Fabian Klute, Irene Parada, and Birgit Vogtenhuber. Crossing-optimal extension of simple drawings. In Nikhil Bansal, Emanuela Merelli, and James Worrell, editors, ICALP 2021, volume 198 of LIPIcs, pages 72:1-72:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2021. URL: https://doi.org/10.4230/LIPIcs.ICALP.2021.72.
  16. Michael R. Garey and David S. Johnson. Crossing number is NP-complete. SIAM J. Algebr. Discrete Methods, 4(3):312-316, September 1983. Google Scholar
  17. Martin Grohe. Computing crossing numbers in quadratic time. J. Comput. Syst. Sci., 68(2):285-302, 2004. URL: https://doi.org/10.1016/j.jcss.2003.07.008.
  18. César Hernández-Vélez, Gelasio Salazar, and Robin Thomas. Nested cycles in large triangulations and crossing-critical graphs. J. Comb. Theory, Ser. B, 102(1):86-92, 2012. Google Scholar
  19. Petr Hliněný. Crossing number is hard for cubic graphs. Journal of Comb. Theory, Ser. B, 96(4):455-471, 2006. URL: https://doi.org/10.1016/j.jctb.2005.09.009.
  20. John Hopcroft and Robert Tarjan. Efficient planarity testing. J. ACM, 21(4):549-568, October 1974. URL: https://doi.org/10.1145/321850.321852.
  21. Vít Jelínek, Jan Kratochvíl, and Ignaz Rutter. A Kuratowski-type theorem for planarity of partially embedded graphs. Computational Geometry, 46(4):466-492, 2013. SoCG 2011. URL: https://doi.org/10.1016/j.comgeo.2012.07.005.
  22. Ken-ichi Kawarabayashi and Bruce Reed. Computing crossing number in linear time. In Proceedings of the Thirty-Ninth Annual ACM Symposium on Theory of Computing, STOC '07, pages 382-390. Association for Computing Machinery, 2007. URL: https://doi.org/10.1145/1250790.1250848.
  23. Kazuo Misue, Peter Eades, Wei Lai, and Kozo Sugiyama. Layout adjustment and the mental map. Journal of Visual Languages and Computing, 6(2):183-210, 1995. URL: https://doi.org/10.1006/jvlc.1995.1010.
  24. Bojan Mohar and Carsten Thomassen. Graphs on Surfaces. Johns Hopkins series in the mathematical sciences. Johns Hopkins University Press, 2001. Google Scholar
  25. Michael J. Pelsmajer, Marcus Schaefer, and Daniel Stefankovic. Crossing numbers of graphs with rotation systems. Algorithmica, 60(3):679-702, 2011. Google Scholar
  26. Robert B. Richter and Carsten Thomassen. Minimal graphs with crossing number at least k. J. Comb. Theory, Ser. B, 58(2):217-224, 1993. Google Scholar
  27. Klaus Wagner. Über eine Eigenschaft der ebenen Komplexe. Mathematische Annalen, 114:570-590, 1937. Google Scholar
  28. Shih Wei-Kuan and Hsu Wen-Lian. A new planarity test. Theoretical Computer Science, 223(1):179-191, 1999. URL: https://doi.org/10.1016/S0304-3975(98)00120-0.
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2024 Schloss Dagstuhl – LZI GmbH Imprint Privacy Contact