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A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth

Authors Sergio Cabello , Alexander Dobler , Gašper Fijavž , Thekla Hamm , Mirko H. Wagner

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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Fixed parameter tractability
Keywords
  • 1-planar
  • crossing type
  • treewidth
  • pathwidth

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Abstract

A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type characterizes the subgraph induced by the four vertices of the crossing edges. Each of the 63 possible nonempty subsets 𝒮 of crossing types gives a recognition problem: does a given graph admit an 𝒮-restricted drawing, that is, a 1-planar drawing where the crossing type of each crossing is in 𝒮?
We show that there is a set 𝒮_bad with three crossing types and the following properties:  
- If 𝒮 contains no crossing type from 𝒮_bad, then the recognition of graphs that admit an 𝒮-restricted drawing is fixed-parameter tractable with respect to the treewidth of the input graph. 
- If 𝒮 contains any crossing type from 𝒮_bad, then it is NP-hard to decide whether a graph has an 𝒮-restricted drawing, even when considering graphs of constant pathwidth.  We also extend this characterization of crossing types to 1-planar straight-line drawings and show the same complexity behaviour parameterized by treewidth.

Cite As Get BibTex

Sergio Cabello, Alexander Dobler, Gašper Fijavž, Thekla Hamm, and Mirko H. Wagner. A Dichotomy for 1-Planarity with Restricted Crossing Types Parameterized by Treewidth. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 16:1-16:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.16

Author Details

Sergio Cabello
  • Faculty of Mathematics and Physics, University of Ljubljana, Slovenia
  • Institute of Mathematics, Physics and Mechanics, Ljubljana, Slovenia
Alexander Dobler
  • TU Wien, Austria
Gašper Fijavž
  • Faculty of Computer and Information Science, University of Ljubljana, Slovenia
Thekla Hamm
  • Department of Mathematics and Computer Science, TU Eindhoven, The Netherlands
Mirko H. Wagner
  • Theoretical Computer Science, Osnabrück University, Germany

Funding

  • Cabello, Sergio: Funded in part by the Slovenian Research and Innovation Agency (P1-0297, N1-0218, N1-0285). Funded in part by the European Union (ERC, KARST, project number 101071836). Views and opinions expressed are however those of the authors only and do not necessarily reflect those of the European Union or the European Research Council. Neither the European Union nor the granting authority can be held responsible for them.
  • Dobler, Alexander: Vienna Science and Technology Fund (WWTF) grant [10.47379/ICT19035].

Acknowledgements

This work was started during the Crossing Number Workshop 2024 that took place in Montserrat, Spain. We thank the organizers and participants for providing a fruitful research environment.

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