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On 2-Layer k-Matching-Planar Graphs

Authors Saeed Odak , Jonathan Rollin , Torben Scheele

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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Computational geometry
  • Theory of computation → Graph algorithms analysis
Keywords
  • pathwidth
  • k-matching-planar graph
  • 2-layer drawings
  • NP-hardness
  • k-planar graph
  • one-sided graph drawing

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Abstract

A graph is k-matching-planar if it admits a drawing in the plane such that, for every edge e, the edges crossing e contain no matching of size greater than k. The class of k-matching-planar graphs generalizes other beyond-planar graph classes, such as k-planar and fan-planar graphs. In a 2-layer drawing of a bipartite graph, the vertices of the two bipartition classes are placed on two parallel horizontal lines and edges are drawn as straight-line segments between them.
We prove that every graph with a 2-layer k-matching-planar drawing has pathwidth at most 2k+1. Moreover, for every k ≥ 0, we construct a graph with a 2-layer k-matching-planar drawing whose pathwidth is 3⌊k/2⌋ + 1. On the algorithmic side, we consider the one-sided recognition problem where a fixed embedding of the vertices on one side is given. We show that this problem is NP-hard. On the other hand, we prove that the problem is fixed-parameter tractable with respect to k. Finally, we prove that the two-sided variant cannot be approximated within any constant factor in polynomial time unless P = NP.

Cite As Get BibTex

Saeed Odak, Jonathan Rollin, and Torben Scheele. On 2-Layer k-Matching-Planar Graphs. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 15:1-15:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.15

Author Details

Saeed Odak
  • Aalto University, Finland
Jonathan Rollin
  • FernUniversität in Hagen, Germany
Torben Scheele
  • TU Dortmund, Germany

Funding

  • Scheele, Torben: Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - project number 550144388.

Acknowledgements

This work was initiated at the Rhein-Ruhr Computational Geometry Workshop RRW 25. We thank the organizers and the participants for the fruitful atmosphere.

References

  1. Patrizio Angelini, Michael A. Bekos, Michael Kaufmann, Philipp Kindermann, and Thomas Schneck. 1-fan-bundle-planar drawings of graphs. Theor. Comput. Sci., 723:23-50, 2018. URL: https://doi.org/10.1016/J.TCS.2018.03.005.
  2. Patrizio Angelini, Giordano Da Lozzo, Giuseppe Di Battista, Fabrizio Frati, and Maurizio Patrignani. 2-level quasi-planarity or how caterpillars climb (SPQR-) trees. In Proceedings of the ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 2779-2798. SIAM, 2021. URL: https://doi.org/10.1137/1.9781611976465.165.
  3. Patrizio Angelini, Giordano Da Lozzo, Henry Förster, and Thomas Schneck. 2-layer k-planar graphs density, crossing lemma, relationships and pathwidth. Comput. J., 67(3):1005-1016, 2024. URL: https://doi.org/10.1093/COMJNL/BXAD038.
  4. Therese Biedl, Steven Chaplick, Michael Kaufmann, Fabrizio Montecchiani, Martin Nöllenburg, and Chrysanthi N. Raftopoulou. Layered fan-planar graph drawings. In Javier Esparza and Daniel Král', editors, MFCS 2020, volume 170 of LIPIcs, pages 14:1-14:13. LIPIcs, 2020. URL: https://doi.org/10.4230/LIPIcs.MFCS.2020.14.
  5. Carla Binucci, Markus Chimani, Walter Didimo, Martin Gronemann, Karsten Klein, Jan Kratochvíl, Fabrizio Montecchiani, and Ioannis G. Tollis. Algorithms and characterizations for 2-layer fan-planarity: From caterpillar to stegosaurus. J. Graph Algorithms Appl., 21(1):81-102, 2017. URL: https://doi.org/10.7155/JGAA.00398.
  6. Franz J. Brandenburg. On fan-crossing graphs. Theor. Comput. Sci., 841:39-49, 2020. URL: https://doi.org/10.1016/J.TCS.2020.07.002.
  7. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  8. Erik D. Demaine, Mohammad Taghi Hajiaghayi, and Ken-ichi Kawarabayashi. Algorithmic graph minor theory: Decomposition, approximation, and coloring. In FOCS 2005, pages 637-646. IEEE Computer Society, 2005. URL: https://doi.org/10.1109/SFCS.2005.14.
  9. Emilio Di Giacomo, Walter Didimo, Peter Eades, and Giuseppe Liotta. 2-layer right angle crossing drawings. Algorithmica, 68(4):954-997, 2014. URL: https://doi.org/10.1007/S00453-012-9706-7.
  10. Walter Didimo, Giuseppe Liotta, and Fabrizio Montecchiani. A survey on graph drawing beyond planarity. ACM Comput. Surv., 52(1):4:1-4:37, 2019. URL: https://doi.org/10.1145/3301281.
  11. Chandan K. Dubey, Uriel Feige, and Walter Unger. Hardness results for approximating the bandwidth. J. Comput. Syst. Sci., 77(1):62-90, 2011. URL: https://doi.org/10.1016/J.JCSS.2010.06.006.
  12. Vida Dujmovic, Michael R. Fellows, Matthew Kitching, Giuseppe Liotta, Catherine McCartin, Naomi Nishimura, Prabhakar Ragde, Frances A. Rosamond, Sue Whitesides, and David R. Wood. On the parameterized complexity of layered graph drawing. Algorithmica, 52(2):267-292, 2008. URL: https://doi.org/10.1007/S00453-007-9151-1.
  13. Peter Eades and Sue Whitesides. Drawing graphs in two layers. Theor. Comput. Sci., 131(2):361-374, 1994. URL: https://doi.org/10.1016/0304-3975(94)90179-1.
  14. Peter Eades and Nicholas C. Wormald. Edge crossings in drawings of bipartite graphs. Algorithmica, 11(4):379-403, 1994. URL: https://doi.org/10.1007/BF01187020.
  15. Oksana Firman, Grzegorz Gutowski, Myroslav Kryven, Yuto Okada, and Alexander Wolff. Bounding the treewidth of outer k-planar graphs via triangulations. In Stefan Felsner and Karsten Klein, editors, 32nd International Symposium on Graph Drawing and Network Visualization, LIPIcs, pages 14:1-14:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2024. URL: https://doi.org/10.4230/LIPIcs.GD.2024.14.
  16. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  17. Michael R. Garey and David S. Johnson. Crossing number is NP-complete. SIAM J. Algebraic Discrete Methods, 4(3):312-316, 1983. URL: https://doi.org/10.1137/0604033.
  18. Panos Giannopoulos, Miriam Goetze, Grzegorz Gutowski, Maarten Löffler, Martin Nöllenburg, Yuto Okada, Jonathan Rollin, Birgit Vogtenhuber, and Alexander Wolff. One-sided local crossing minimization. CoRR, 2510.00331, 2026. URL: https://doi.org/10.48550/arXiv.2510.00331.
  19. Alexander Grigoriev and Hans L Bodlaender. Algorithms for graphs embeddable with few crossings per edge. Algorithmica, 49(1):1-11, 2007. URL: https://doi.org/10.1007/S00453-007-0010-X.
  20. Patrick Healy and Nikola S. Nikolov. Hierarchical drawing algorithms. In Roberto Tamassia, editor, Handbook on Graph Drawing and Visualization, pages 409-453. Chapman and Hall/CRC, 2013. URL: https://doi.org/10.1201/b15385.
  21. Kevin Hendrey, Nikolai Karol, and David R. Wood. Structure of k-matching-planar graphs. CoRR, abs/2507.22395, 2025. URL: https://doi.org/10.48550/arXiv.2507.22395.
  22. Petr Hliněný. Crossing-number critical graphs have bounded path-width. J. Comb. Theory B, 88(2):347-367, 2003. URL: https://doi.org/10.1016/S0095-8956(03)00037-6.
  23. Michael Hoffmann, Chih-Hung Liu, Meghana M Reddy, and Csaba D. Tóth. Simple topological drawings of k-planar graphs. In International Symposium on Graph Drawing and Network Visualization, pages 390-402. Springer, 2020. URL: https://doi.org/10.1007/978-3-030-68766-3_31.
  24. Seok-Hee Hong and Takeshi Tokuyama. Beyond planar graphs. Springer, 2020. Google Scholar
  25. Jisu Jeong, Sigve Hortemo Sæther, and Jan Arne Telle. Maximum matching width: New characterizations and a fast algorithm for dominating set. Discret. Appl. Math., 248:114-124, 2018. URL: https://doi.org/10.1016/J.DAM.2017.09.019.
  26. Michael Kaufmann and Torsten Ueckerdt. The density of fan-planar graphs. Electron. J. Comb., 29(1), 2022. URL: https://doi.org/10.37236/10521.
  27. Daniel J Kleitman. The crossing number of K_5, n. Journal of Combinatorial Theory, 9(4):315-323, 1970. Google Scholar
  28. Yasuaki Kobayashi and Yuto Okada. 2-layer fan-planarity in polynomial time. CoRR, abs/2508.17349, 2025. URL: https://doi.org/10.48550/arXiv.2508.17349.
  29. Yasuaki Kobayashi, Yuto Okada, and Alexander Wolff. Recognizing 2-layer and outer k-planar graphs. In Oswin Aichholzer and Haitao Wang, editors, SoCG 2025, volume 332 of LIPIcs, pages 65:1-65:16. LIPIcs, 2025. URL: https://doi.org/10.4230/LIPIcs.SOCG.2025.65.
  30. Laura Merker, Lena Scherzer, Samuel Schneider, and Torsten Ueckerdt. Intersection graphs with and without product structure. In Stefan Felsner and Karsten Klein, editors, GD 2024, volume 320 of LIPIcs, pages 23:1-23:19. LIPIcs, 2024. URL: https://doi.org/10.4230/LIPIcs.GD.2024.23.
  31. Yuto Okada. Pathwidth of 2-layer k-planar graphs. CoRR, 2507.21864, 2025. URL: https://doi.org/10.48550/arXiv.2507.21864.
  32. János Pach and Géza Tóth. Graphs drawn with few crossings per edge. Combinatorica, 17(3):427-439, 1997. URL: https://doi.org/10.1007/BF01215922.
  33. Neil Robertson and Paul D. Seymour. Graph minors. i. excluding a forest. J. Comb. Theory B, 35(1):39-61, 1983. URL: https://doi.org/10.1016/0095-8956(83)90079-5.
  34. Matthew Suderman. Pathwidth and layered drawings of trees. Int. J. Comput. Geom. Appl., 14(3):203-225, 2004. URL: https://doi.org/10.1142/S0218195904001433.
  35. David R. Wood. 2-layer graph drawings with bounded pathwidth. J. Graph Algorithms Appl., 27(9):843-851, 2023. URL: https://doi.org/10.7155/JGAA.00647.
  36. Meirav Zehavi. Parameterized analysis and crossing minimization problems. Comput. Sci. Rev., 45:100490, 2022. URL: https://doi.org/10.1016/J.COSREV.2022.100490.
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