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Bounding the Treewidth of Outer k-Planar Graphs via Triangulations

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

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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

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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

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