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Parameterized Algorithms for Beyond-Planar Crossing Numbers

Authors Miriam Münch , Ignaz Rutter

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LIPIcs.GD.2024.25.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Fixed parameter tractability
Keywords
  • FPT
  • Beyond-planarity
  • Crossing-number
  • Crossing Patterns

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Abstract

Beyond-planar graph classes are usually defined via forbidden configurations or patterns in a drawing. In this paper, we formalize these concepts on a combinatorial level and show that, for any fixed family ℱ of crossing patterns, deciding whether a given graph G admits a drawing that avoids all patterns in F and that has at most c crossings is FPT w.r.t. c. In particular, we show that for any fixed k, deciding whether a graph is k-planar, k-quasi-planar, fan-crossing, fan-crossing-free or min-k-planar, respectively, is FPT with respect to the corresponding beyond-planar crossing number.

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Miriam Münch and Ignaz Rutter. Parameterized Algorithms for Beyond-Planar Crossing Numbers. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 25:1-25:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.25

Author Details

Miriam Münch
  • Faculty of Computer Science and Mathematics, University of Passau, Germany
Ignaz Rutter
  • Faculty of Computer Science and Mathematics, University of Passau, Germany

Funding

Funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 541433306.

References

  1. Pankaj K Agarwal, Boris Aronov, János Pach, Richard Pollack, and Micha Sharir. Quasi-planar graphs have a linear number of edges. Combinatorica, 17(1):1-9, 1997. URL: https://doi.org/10.1007/BF01196127.
  2. Patrizio Angelini, Michael A Bekos, Franz J Brandenburg, Giordano Da Lozzo, Giuseppe Di Battista, Walter Didimo, Giuseppe Liotta, Fabrizio Montecchiani, and Ignaz Rutter. On the relationship between k-planar and k-quasi-planar graphs. In Graph-Theoretic Concepts in Computer Science: 43rd International Workshop, WG 2017, pages 59-74. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-68705-6_5.
  3. 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.
  4. Sang Won Bae, Jean-François Baffier, Jinhee Chun, Peter Eades, Kord Eickmeyer, Luca Grilli, Seok-Hee Hong, Matias Korman, Fabrizio Montecchiani, Ignaz Rutter, and Csaba D. Tóth. Gap-planar graphs. In Graph Drawing and Network Visualization - 25th International Symposium, GD 2017, volume 10692 of Lecture Notes in Computer Science, pages 531-545. Springer, 2017. URL: https://doi.org/10.1007/978-3-319-73915-1_41.
  5. Sang Won Bae, Jean-Francois Baffier, Jinhee Chun, Peter Eades, Kord Eickmeyer, Luca Grilli, Seok-Hee Hong, Matias Korman, Fabrizio Montecchiani, Ignaz Rutter, and Csaba D. Tóth. Gap-planar graphs. Theoretical Computer Science, 745:36-52, 2018. URL: https://doi.org/10.1016/J.TCS.2018.05.029.
  6. Michael J. Bannister, Sergio Cabello, and David Eppstein. Parameterized complexity of 1-planarity. Journal of Graph Algorithms and Applications, 22(1):23-49, 2018. URL: https://doi.org/10.7155/jgaa.00457.
  7. Carla Binucci, Aaron Büngener, Giuseppe Di Battista, Walter Didimo, Vida Dujmovic, Seok-Hee Hong, Michael Kaufmann, Giuseppe Liotta, Pat Morin, and Alessandra Tappini. Min-k-planar drawings of graphs. In Graph Drawing and Network Visualization - 31st International Symposium, GD 2023, volume 14465 of Lecture Notes in Computer Science, pages 39-52. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-49272-3_3.
  8. Carla Binucci, Aaron Büngener, Giuseppe Di Battista, Walter Didimo, Vida Dujmovic, Seok-Hee Hong, Michael Kaufmann, Giuseppe Liotta, Pat Morin, and Alessandra Tappini. Min-k-planar drawings of graphs. J. Graph Algorithms Appl., 28(2):1-35, 2024. URL: https://doi.org/10.7155/JGAA.V28I2.2925.
  9. Carla Binucci, Emilio Di Giacomo, Walter Didimo, Fabrizio Montecchiani, Maurizio Patrignani, Antonios Symvonis, and Ioannis G Tollis. Fan-planarity: Properties and complexity. Theoretical Computer Science, 589:76-86, 2015. URL: https://www.sciencedirect.com/science/article/pii/S0304397515003515, URL: https://doi.org/10.1016/J.TCS.2015.04.020.
  10. Thomas Bläsius, Stephen G. Kobourov, and Ignaz Rutter. Simultaneous embedding of planar graphs. In Roberto Tamassia, editor, Handbook on Graph Drawing and Visualization, pages 349-381. Chapman and Hall/CRC, 2013. Google Scholar
  11. Hans L. Bodlaender. A linear-time algorithm for finding tree-decompositions of small treewidth. SIAM J. Comput., 25(6):1305-1317, 1996. URL: https://doi.org/10.1137/S0097539793251219.
  12. Franz J Brandenburg. On 4-map graphs and 1-planar graphs and their recognition problem. arXiv preprint, 2015. URL: https://arxiv.org/abs/1509.03447.
  13. Franz J. Brandenburg. Recognizing optimal 1-planar graphs in linear time. Algorithmica, 80(1):1-28, 2018. URL: https://doi.org/10.1007/s00453-016-0226-8.
  14. Franz J. Brandenburg. On fan-crossing graphs. Theoretical Computer Science, 841:39-49, 2020. URL: https://doi.org/10.1016/j.tcs.2020.07.002.
  15. Franz J Brandenburg. Fan-crossing free graphs and their relationship to other beyond-planar graphs. Theoretical Computer Science, 867:85-100, 2021. URL: https://doi.org/10.1016/J.TCS.2021.03.031.
  16. Sergio Cabello and Bojan Mohar. Adding one edge to planar graphs makes crossing number and 1-planarity hard. SIAM Journal on Computing, 42(5):1803-1829, 2013. URL: https://doi.org/10.1137/120872310.
  17. Zhi-Zhong Chen, Michelangelo Grigni, and Christos H Papadimitriou. Recognizing hole-free 4-map graphs in cubic time. Algorithmica, 45(2):227-262, 2006. URL: https://link.springer.com/content/pdf/10.1007/s00453-005-1184-8.pdf, URL: https://doi.org/10.1007/S00453-005-1184-8.
  18. Otfried Cheong, Sariel Har-Peled, Heuna Kim, and Hyo-Sil Kim. On the number of edges of fan-crossing free graphs. Algorithmica, 73(4):673-695, 2015. URL: https://doi.org/10.1007/S00453-014-9935-Z.
  19. Markus Chimani, Max Ilsen, and Tilo Wiedera. Star-struck by fixed embeddings: Modern crossing number heuristics. In Graph Drawing and Network Visualization - 29st International Symposium, GD 2021, volume 12868 of Lecture Notes in Computer Science, pages 41-56. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-92931-2_3.
  20. Julia Chuzhoy and Zihan Tan. A subpolynomial approximation algorithm for graph crossing number in low-degree graphs. In Proceedings of the 54th Annual ACM SIGACT Symposium on Theory of Computing, pages 303-316, 2022. URL: https://doi.org/10.1145/3519935.3519984.
  21. Bruno Courcelle. The monadic second-order theory of graphs I: Recognizable sets of finite graphs. Information and Computation, 1990. Google Scholar
  22. Walter Didimo, Giuseppe Liotta, and Fabrizio Montecchiani. A survey on graph drawing beyond planarity. ACM Computing Surveys, 52(1):4:1-4:37, 2019. URL: https://doi.org/10.1145/3301281.
  23. Henry Förster, Michael Kaufmann, and Chrysanthi N Raftopoulou. Recognizing and embedding simple optimal 2-planar graphs. In Graph Drawing and Network Visualization - 29st International Symposium, GD 2021, pages 87-100. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-92931-2_6.
  24. Michael R Garey and David S Johnson. Crossing number is NP-complete. SIAM Journal on Algebraic Discrete Methods, 4(3):312-316, 1983. Google Scholar
  25. Martin Grohe. Computing crossing numbers in quadratic time. Journal of Computer and System Sciences, 68(2):285-302, 2004. URL: https://doi.org/10.1016/j.jcss.2003.07.008.
  26. Thekla Hamm and Petr Hlinený. Parameterised partially-predrawn crossing number. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry (SoCG '22), volume 224 of LIPIcs, pages 46:1-46:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.46.
  27. Petr Hliněnỳ and Markus Chimani. Approximating the crossing number of graphs embeddable in any orientable surface. In Proceedings of the Twenty-First Annual ACM-SIAM Symposium on Discrete Algorithms, pages 918-927. SIAM, 2010. Google Scholar
  28. Seok-Hee Hong, Peter Eades, Naoki Katoh, Giuseppe Liotta, Pascal Schweitzer, and Yusuke Suzuki. A linear-time algorithm for testing outer-1-planarity. Algorithmica, 72(4):1033-1054, 2015. URL: https://doi.org/10.1007/s00453-014-9890-8.
  29. Seok-Hee Hong and Takeshi Tokuyama, editors. Beyond Planar Graphs. Springer Singapore, 2020. URL: https://doi.org/10.1007/978-981-15-6533-5.
  30. Weidong Huang, Seok-Hee Hong, and Peter Eades. Effects of crossing angles. In IEEE VGTC Pacific Visualization Symposium 2008, PacificVis 2008, Kyoto, Japan, March 5-7, 2008, pages 41-46. IEEE, 2008. URL: https://doi.org/10.1109/PACIFICVIS.2008.4475457.
  31. Michael Kaufmann and Torsten Ueckerdt. The density of fan-planar graphs. The Electronic Journal of Combinatorics, 29(1), 2022. URL: https://doi.org/10.37236/10521.
  32. Ken-ichi Kawarabayashi and Buce Reed. Computing crossing number in linear time. In Proceedings of the thirty-ninth annual ACM symposium on Theory of computing, pages 382-390, 2007. URL: https://doi.org/10.1145/1250790.1250848.
  33. Stephen G Kobourov, Sergey Pupyrev, and Bahador Saket. Are crossings important for drawing large graphs? In Graph Drawing: 22nd International Symposium, GD 2014, pages 234-245. Springer, 2014. URL: https://doi.org/10.1007/978-3-662-45803-7_20.
  34. Vladimir P Korzhik and Bojan Mohar. Minimal obstructions for 1-immersions and hardness of 1-planarity testing. Journal of Graph Theory, 72(1):30-71, 2013. URL: https://onlinelibrary.wiley.com/doi/epdf/10.1002/jgt.21630, URL: https://doi.org/10.1002/JGT.21630.
  35. János Pach and Géza Tóth. Graphs drawn with few crossings per edge. Combinatorica, 17(3):427-439, 1997. URL: https://doi.org/10.1007/3-540-62495-3_59.
  36. Michael J Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Crossing numbers and parameterized complexity. In Graph Drawing: 15th International Symposium, GD 2007., pages 31-36. Springer, 2008. Google Scholar
  37. Ljubomir Perkovic and Bruce A. Reed. An improved algorithm for finding tree decompositions of small width. Int. J. Found. Comput. Sci., 11(3):365-371, 2000. URL: https://doi.org/10.1142/S0129054100000247.
  38. H. C. Purchase, R. F. Cohen, and M. I. James. An experimental study of the basis for graph drawing algorithms. ACM Journal of Experimental Algorithmics, 2:4, January 1997. URL: https://doi.org/10.1145/264216.264222.
  39. Helen C. Purchase. Which aesthetic has the greatest effect on human understanding? In Giuseppe Di Battista, editor, Proceedings 5th International Symposium on Graph Drawing (GD '97), volume 1353 of Lecture Notes in Computer Science, pages 248-261. Springer, 1997. URL: https://doi.org/10.1007/3-540-63938-1_67.
  40. Helen C. Purchase, Jo-Anne Allder, and David A. Carrington. Graph layout aesthetics in UML diagrams: User preferences. Journal of Graph Algorithms and Applications, 6(3):255-279, 2002. URL: https://doi.org/10.7155/jgaa.00054.
  41. Neil Robertson and Paul D. Seymour. Graph minors .xiii. the disjoint paths problem. J. Comb. Theory, Ser. B, 63(1):65-110, 1995. URL: https://doi.org/10.1006/jctb.1995.1006.
  42. Ignaz Rutter. Simultaneous embedding. In Beyond Planar Graphs: Communications of NII Shonan Meetings, pages 237-265. Springer, 2020. URL: https://doi.org/10.1007/978-981-15-6533-5_13.
  43. Carsten Thomassen. A simpler proof of the excluded minor theorem for higher surfaces. J. Comb. Theory, Ser. B, 70(2):306-311, 1997. URL: https://doi.org/10.1006/jctb.1997.1761.
  44. John C Urschel and Jake Wellens. Testing gap k-planarity is NP-complete. Information Processing Letters, 169:106083, 2021. URL: https://doi.org/10.1016/J.IPL.2020.106083.
  45. Nathan van Beusekom, Irene Parada, and Bettina Speckmann. Crossing numbers of beyond-planar graphs revisited. In 29th International Symposium on Graph Drawing and Network Visualization (GD '21), volume 12868, page 447. Springer Nature, 2021. Google Scholar
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