Document Open Access Logo

Bounding the Treewidth of Outer k-Planar Graphs via Triangulations

Authors Oksana Firman , Grzegorz Gutowski , Myroslav Kryven , Yuto Okada , Alexander Wolff

PDF

File

Thumbnail PDF
PDF
LIPIcs.GD.2024.14.pdf
  • Filesize: 1.11 MB
  • 17 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph theory
Keywords
  • treewidth
  • outerplanar graphs
  • outer k-planar graphs
  • outer min-k-planar graphs
  • cop number
  • separation number

Metrics

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

Abstract

The treewidth is a structural parameter that measures the tree-likeness of a graph. Many algorithmic and combinatorial results are expressed in terms of the treewidth. In this paper, we study the treewidth of outer k-planar graphs, that is, graphs that admit a straight-line drawing where all the vertices lie on a circle, and every edge is crossed by at most k other edges.
Wood and Telle [New York J. Math., 2007] showed that every outer k-planar graph has treewidth at most 3k + 11 using so-called planar decompositions, and later, Auer et al. [Algorithmica, 2016] proved that the treewidth of outer 1-planar graphs is at most 3, which is tight.
In this paper, we improve the general upper bound to 1.5k + 2 and give a tight bound of 4 for k = 2. We also establish a lower bound: we show that, for every even k, there is an outer k-planar graph with treewidth k+2. Our new bound immediately implies a better bound on the cop number, which answers an open question of Durocher et al. [GD 2023] in the affirmative.
Our treewidth bound relies on a new and simple triangulation method for outer k-planar graphs that yields few crossings with graph edges per edge of the triangulation. Our method also enables us to obtain a tight upper bound of k + 2 for the separation number of outer k-planar graphs, improving an upper bound of 2k + 3 by Chaplick et al. [GD 2017]. We also consider outer min-k-planar graphs, a generalization of outer k-planar graphs, where we achieve smaller improvements.

Cite As Get BibTex

Oksana Firman, Grzegorz Gutowski, Myroslav Kryven, Yuto Okada, and Alexander Wolff. Bounding the Treewidth of Outer k-Planar Graphs via Triangulations. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 14:1-14:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.14

Author Details

Oksana Firman
  • Universität Würzburg, Germany
Grzegorz Gutowski
  • Institute of Theoretical Computer Science, Faculty of Mathematics and Computer Science, Jagiellonian University, Kraków, Poland
Myroslav Kryven
  • University of Manitoba, Canada
Yuto Okada
  • Nagoya University, Japan
Alexander Wolff
  • Universität Würzburg, Germany

Funding

  • Firman, Oksana: Supported by DFG grant Wo758/9-1.
  • Gutowski, Grzegorz: The research cooperation was funded by the program Excellence Initiative - Research University at the Jagiellonian University in Kraków.
  • Okada, Yuto: Supported by JST SPRING, Grant Number JPMJSP2125 and JSPS KAKENHI, Grant Number JP22H00513 (Hirotaka Ono).

Acknowledgements

We thank Yota Otachi for suggestions that helped us to improve the lower bound. We also thank Hirotaka Ono for supporting our work.

References

  1. Ivan Aidun, Frances Dean, Ralph Morrison, Teresa Yu, and Julie Yuan. Treewidth and gonality of glued grid graphs. Discrete Applied Mathematics, 279:1-11, 2020. URL: https://doi.org/10.1016/j.dam.2019.10.024.
  2. Patrizio Angelini, Giordano Da Lozzo, Henry Förster, and Thomas Schneck. 2-layer k-planar graphs: Density, crossing lemma, relationships and pathwidth. Computer Journal, 67(3):1005-1016, 2024. URL: https://doi.org/10.1093/comjnl/bxad038.
  3. Patrizio Angelini, Giordano Da Lozzo, Henry Förster, and Thomas Schneck. 2-layer k-planar graphs - density, crossing lemma, relationships, and pathwidth. In David Auber and Pavel Valtr, editors, 28th International Symposium on Graph Drawing and Network Visualization (GD 2020), volume 12590 of Lecture Notes in Computer Science, pages 403-419. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-68766-3_32.
  4. Christopher Auer, Christian Bachmaier, Franz J. Brandenburg, Andreas Gleißner, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber. Recognizing outer 1-planar graphs in linear time. In Stephen K. Wismath and Alexander Wolff, editors, 21st International Symposium on Graph Drawing (GD 2013), volume 8242 of Lecture Notes in Computer Science, pages 107-118. Springer, 2013. URL: https://doi.org/10.1007/978-3-319-03841-4_10.
  5. Christopher Auer, Christian Bachmaier, Franz J. Brandenburg, Andreas Gleißner, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber. Outer 1-planar graphs. Algorithmica, 74(4):1293-1320, 2016. URL: https://doi.org/10.1007/s00453-015-0002-1.
  6. Christopher Auer, Christian Bachmaier, Franz J. Brandenburg, Andreas Gleißner, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber. Correction to: Outer 1-planar graphs. Algorithmica, 83(11):3534-3535, 2021. URL: https://doi.org/10.1007/S00453-021-00874-Z.
  7. Michael J. Bannister and David Eppstein. Crossing minimization for 1-page and 2-page drawings of graphs with bounded treewidth. In Christian A. Duncan and Antonios Symvonis, editors, 22nd International Symposium on Graph Drawing (GD 2014), volume 8871 of Lecture Notes in Computer Science, pages 210-221. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-45803-7_18.
  8. Michael J. Bannister and David Eppstein. Crossing minimization for 1-page and 2-page drawings of graphs with bounded treewidth. Journal of Graph Algorithms and Applications, 22(4):577-606, 2018. URL: https://doi.org/10.7155/jgaa.00479.
  9. Hans L. Bodlaender. A partial k-arboretum of graphs with bounded treewidth. Theoretical Computer Science, 209(1-2):1-45, 1998. URL: https://doi.org/10.1016/S0304-3975(97)00228-4.
  10. Steven Chaplick, Myroslav Kryven, Giuseppe Liotta, Andre Löffler, and Alexander Wolff. Beyond outerplanarity. In Fabrizio Frati and Kwan-Liu Ma, editors, 25th International Symposium on Graph Drawing and Network Visualization (GD 2017), volume 10692 of Lecture Notes in Computer Science, pages 546-559. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-73915-1_42.
  11. Steven Chaplick, Thomas C. van Dijk, Myroslav Kryven, Ji-won Park, Alexander Ravsky, and Alexander Wolff. Bundled crossings revisited. In Daniel Archambault and Csaba D. Tóth, editors, 27th International Symposium on Graph Drawing and Network Visualization (GD 2019), volume 11904 of Lecture Notes in Computer Science, pages 63-77. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-35802-0_5.
  12. Steven Chaplick, Thomas C. van Dijk, Myroslav Kryven, Ji-won Park, Alexander Ravsky, and Alexander Wolff. Bundled crossings revisited. Journal of Graph Algorithms and Applications, 24(4):621-655, 2020. URL: https://doi.org/10.7155/jgaa.00535.
  13. Bruno Courcelle. The monadic second-order logic of graphs. I. Recognizable sets of finite graphs. Information and Computation, 85(1):12-75, 1990. URL: https://doi.org/10.1016/0890-5401(90)90043-H.
  14. Stephane Durocher, Shahin Kamali, Myroslav Kryven, Fengyi Liu, Amirhossein Mashghdoust, Avery Miller, Pouria Zamani Nezhad, Ikaro Penha Costa, and Timothy Zapp. Cops and robbers on 1-planar graphs. In Michael A. Bekos and Markus Chimani, editors, 31st International Symposium on Graph Drawing and Network Visualization (GD 2023), volume 14466 of Lecture Notes in Computer Science, pages 3-17. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-49275-4_1.
  15. Zdeněk Dvořák and Sergey Norin. Treewidth of graphs with balanced separations. Journal of Combinatorial Theory, Series B, 137:137-144, 2019. URL: https://doi.org/10.1016/j.jctb.2018.12.007.
  16. Gwenaël Joret, Marcin Kamiński, and Dirk Oliver Theis. The cops and robber game on graphs with forbidden (induced) subgraphs. Contributions to Discrete Mathematics, 5(2):40-51, 2010. URL: http://cdm.ucalgary.ca/cdm/index.php/cdm/article/view/154, URL: https://doi.org/10.11575/cdm.v5i2.62032.
  17. Nancy G. Kinnersley and Michael A. Langston. Obstruction set isolation for the gate matrix layout problem. Discrete Applied Mathematics, 54(2-3):169-213, 1994. URL: https://doi.org/10.1016/0166-218X(94)90021-3.
  18. Andrzej Proskurowski and Jan Arne Telle. Classes of graphs with restricted interval models. Discrete Mathematics & Theoretical Computer Science, 3(4):167-176, 1999. URL: https://doi.org/10.46298/dmtcs.263.
  19. Marcus Schaefer. The graph crossing number and its variants: A survey. Electronic Journal of Combinatorics, DS21(version 8), 2024. URL: https://doi.org/10.37236/2713.
  20. David R. Wood and Jan Arne Telle. Planar decompositions and the crossing number of graphs with an excluded minor. New York Journal of Mathematics, 13:117-146, 2007. URL: https://nyjm.albany.edu/j/2007/13-8.html.
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
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact