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Structural Parameterizations of k-Planarity

Authors Tatsuya Gima , Yasuaki Kobayashi , Yuto Okada

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ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • 1-planar graphs
  • local crossing number
  • beyond planarity
  • parameterized complexity
  • kernelization

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Abstract

The concept of k-planarity is extensively studied in the context of Beyond Planarity. A graph is k-planar if it admits a drawing in the plane in which each edge is crossed at most k times. The local crossing number of a graph is the minimum integer k such that it is k-planar. The problem of determining whether an input graph is 1-planar is known to be NP-complete even for near-planar graphs [Cabello and Mohar, SIAM J. Comput. 2013], that is, the graphs obtained from planar graphs by adding a single edge. Moreover, the local crossing number is hard to approximate within a factor 2 - ε for any ε > 0 [Urschel and Wellens, IPL 2021]. To address this computational intractability, Bannister, Cabello, and Eppstein [JGAA 2018] investigated the parameterized complexity of the case of k = 1, particularly focusing on structural parameterizations on input graphs, such as treedepth, vertex cover number, and feedback edge number. In this paper, we extend their approach by considering the general case k ≥ 1 and give (tight) parameterized upper and lower bound results. In particular, we strengthen the aforementioned lower bound results to subclasses of constant-treewidth graphs: we show that testing 1-planarity is NP-complete even for near-planar graphs with feedback vertex set number at most 3 and pathwidth at most 4, and the local crossing number is hard to approximate within any constant factor for graphs with feedback vertex set number at most 2.

Cite As Get BibTex

Tatsuya Gima, Yasuaki Kobayashi, and Yuto Okada. Structural Parameterizations of k-Planarity. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 16:1-16:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.16

Author Details

Tatsuya Gima
  • Hokkaido University, Sapporo, Japan
Yasuaki Kobayashi
  • Hokkaido University, Sapporo, Japan
Yuto Okada
  • Nagoya University, Japan

Funding

  • Gima, Tatsuya: JSPS KAKENHI Grant Number JP24K23847 and JP25K03077.
  • Kobayashi, Yasuaki: JSPS KAKENHI Grant Numbers JP23K28034, JP24H00686, and JP24H00697.
  • Okada, Yuto: Supported by JST SPRING, Grant Number JPMJSP2125.

Acknowledgements

We would like to thank Alexander Wolff for valuable discussions in Würzburg and anonymous reviewers for insightful comments.

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