Skip to main content
Springer Nature Link
Log in

Pathways to Tractability for Geometric Thickness

  • Conference paper
  • First Online:
SOFSEM 2025: Theory and Practice of Computer Science (SOFSEM 2025)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 15538))

  • 26 Accesses

Abstract

We study the classical problem of computing geometric thickness, i.e., finding a straight-line drawing of an input graph and a partition of its edges into as few parts as possible so that each part is crossing-free. Since the problem is NP-hard, we investigate its tractability through the lens of parameterized complexity. As our first set of contributions, we provide two fixed-parameter algorithms which utilize well-studied parameters of the input graph, notably the vertex cover and feedback edge numbers. Since parameterizing by the thickness itself does not yield tractability and the use of other structural parameters remains open due to general challenges identified in previous works, as our second set of contributions, we propose a different pathway to tractability for the problem: extension of partial solutions. In particular, we establish a full characterization of the problem’s parameterized complexity in the extension setting depending on whether we parameterize by the number of missing vertices, edges, or both.

All authors acknowledge support from the Vienna Science and Technology Fund (WWTF) [10.47379/ICT22029]. Robert Ganian and Alexander Firbas also acknowledge support from the Austrian Science Fund (FWF) [10.55776/Y1329].

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Angelini, P., et al.: Testing planarity of partially embedded graphs. ACM Trans. Algorithms 11(4), 32:1–32:42 (2015). https://doi.org/10.1145/2629341

  2. Balabán, J., Ganian, R., Rocton, M.: Computing twin-width parameterized by the feedback edge number. In: Beyersdorff, O., Kanté, M.M., Kupferman, O., Lokshtanov, D. (eds.) 41st International Symposium on Theoretical Aspects of Computer Science, STACS 2024, March 12-14, 2024, Clermont-Ferrand, France. LIPIcs, vol. 289, pp. 7:1–7:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2024). https://doi.org/10.4230/LIPICS.STACS.2024.7

  3. Balko, M., Chaplick, S., Ganian, R., Gupta, S., Hoffmann, M., Valtr, P., Wolff, A.: Bounding and computing obstacle numbers of graphs. In: Chechik, S., Navarro, G., Rotenberg, E., Herman, G. (eds.) 30th Annual European Symposium on Algorithms, ESA 2022, September 5-9, 2022, Berlin/Potsdam, Germany. LIPIcs, vol. 244, pp. 11:1–11:13. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022). https://doi.org/10.4230/LIPICS.ESA.2022.11

  4. Bannister, M.J., Cabello, S., Eppstein, D.: Parameterized complexity of 1-planarity. In: Dehne, F., Solis-Oba, R., Sack, J.R. (eds.) Algorithms and Data Structures, pp. 97–108. Springer Berlin Heidelberg, Berlin, Heidelberg (2013). https://doi.org/10.1007/978-3-642-40104-6_9

  5. Bannister, M.J., Cabello, S., Eppstein, D.: Parameterized complexity of 1-planarity. J. Graph Algorithms Appl. 22(1), 23–49 (2018). https://doi.org/10.7155/JGAA.00457

    Article  MathSciNet  MATH  Google Scholar 

  6. Beineke, L.W., Harary, F.: The thickness of the complete graph. Can. J. Math. 17, 850–859 (1965)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bhore, S., Ganian, R., Khazaliya, L., Montecchiani, F., Nöllenburg, M.: Extending orthogonal planar graph drawings is fixed-parameter tractable. In: Chambers, E.W., Gudmundsson, J. (eds.) 39th International Symposium on Computational Geometry, SoCG 2023, June 12-15, 2023, Dallas, Texas, USA. LIPIcs, vol. 258, pp. 18:1–18:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023). https://doi.org/10.4230/LIPICS.SOCG.2023.18

  8. Bhore, S., Ganian, R., Montecchiani, F., Nöllenburg, M.: Parameterized algorithms for book embedding problems. J. Graph Algorithms Appl. 24(4), 603–620 (2020). https://doi.org/10.7155/JGAA.00526

    Article  MathSciNet  MATH  Google Scholar 

  9. Bhore, S., Ganian, R., Montecchiani, F., Nöllenburg, M.: Parameterized algorithms for queue layouts. J. Graph Algorithms Appl. 26(3), 335–352 (2022). https://doi.org/10.7155/JGAA.00597

    Article  MathSciNet  MATH  Google Scholar 

  10. Binucci, C., et al.: On the complexity of the storyplan problem. J. Comput. Syst. Sci. 139, 103466 (2024). https://doi.org/10.1016/J.JCSS.2023.103466

    Article  MathSciNet  MATH  Google Scholar 

  11. Brand, C., Ganian, R., Röder, S., Schager, F.: Fixed-parameter algorithms for computing RAC drawings of graphs. In: Bekos, M.A., Chimani, M. (eds.) Graph Drawing and Network Visualization - 31st International Symposium, GD 2023, Isola delle Femmine, Palermo, Italy, September 20-22, 2023, Revised Selected Papers, Part II. Lecture Notes in Computer Science, vol. 14466, pp. 66–81. Springer (2023). https://doi.org/10.1007/978-3-031-49275-4_5

  12. Brandenburg, F., Eppstein, D., Goodrich, M.T., Kobourov, S., Liotta, G., Mutzel, P.: Selected open problems in graph drawing. In: Liotta, G. (ed.) GD 2003. LNCS, vol. 2912, pp. 515–539. Springer, Heidelberg (2004). https://doi.org/10.1007/978-3-540-24595-7_55

    Chapter  MATH  Google Scholar 

  13. Cheong, O., Pfister, M., Schlipf, L.: The thickness of fan-planar graphs is at most three. In: Angelini, P., von Hanxleden, R. (eds.) Graph Drawing and Network Visualization - 30th International Symposium, GD 2022, Tokyo, Japan, September 13-16, 2022, Revised Selected Papers. Lecture Notes in Computer Science, vol. 13764, pp. 247–260. Springer (2022). https://doi.org/10.1007/978-3-031-22203-0_18

  14. Courcelle, B.: The monadic second-order logic of graphs. i. recognizable sets of finite graphs. Inf. Comput. 85(1), 12–75 (1990). https://doi.org/10.1016/0890-5401(90)90043-H

  15. Cygan, M., et al.: Parameterized Algorithms. Springer (2015). https://doi.org/10.1007/978-3-319-21275-3

  16. Depian, T., Fink, S.D., Firbas, A., Ganian, R., Nöllenburg, M.: Pathways to tractability for geometric thickness (2024). https://arxiv.org/abs/2411.15864

  17. Depian, T., Fink, S.D., Ganian, R., Nöllenburg, M.: The parameterized complexity of extending stack layouts. In: Proceedings of the 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Vienna, Austria (2024). https://doi.org/10.4230/LIPIcs.GD.2024.12

  18. Diestel, R.: Graph Theory, 4th Edition, Graduate texts in mathematics, vol. 173. Springer (2012)

    Google Scholar 

  19. Dillencourt, M.B., Eppstein, D., Hirschberg, D.S.: Geometric thickness of complete graphs. J. Graph Algorithms Appl. 4(3), 5–17 (2000). https://doi.org/10.7155/JGAA.00023

    Article  MathSciNet  MATH  Google Scholar 

  20. Downey, R.G., Fellows, M.R.: Fundamentals of Parameterized Complexity. Texts in Computer Science, Springer (2013). https://doi.org/10.1007/978-1-4471-5559-1

  21. Dujmović, V., Wood, D.R.: Graph treewidth and geometric thickness parameters. Discret. Comput. Geom. 37(4), 641–670 (2007). https://doi.org/10.1007/S00454-007-1318-7

    Article  MathSciNet  MATH  Google Scholar 

  22. Durocher, S., Gethner, E., Mondal, D.: Thickness and colorability of geometric graphs. Comput. Geom. 56, 1–18 (2016). https://doi.org/10.1016/J.COMGEO.2016.03.003

    Article  MathSciNet  MATH  Google Scholar 

  23. Durocher, S., Mondal, D.: Relating graph thickness to planar layers and bend complexity. SIAM J. Discret. Math. 32(4), 2703–2719 (2018). https://doi.org/10.1137/16M1110042

    Article  MathSciNet  MATH  Google Scholar 

  24. Eiben, E., Ganian, R., Hamm, T., Klute, F., Nöllenburg, M.: Extending nearly complete 1-planar drawings in polynomial time. In: Esparza, J., Král’, D. (eds.) 45th International Symposium on Mathematical Foundations of Computer Science, MFCS 2020, August 24-28, 2020, Prague, Czech Republic. LIPIcs, vol. 170, pp. 31:1–31:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPICS.MFCS.2020.31

  25. Eiben, E., Ganian, R., Hamm, T., Klute, F., Nöllenburg, M.: Extending partial 1-planar drawings. In: Czumaj, A., Dawar, A., Merelli, E. (eds.) 47th International Colloquium on Automata, Languages, and Programming, ICALP 2020, July 8-11, 2020, Saarbrücken, Germany (Virtual Conference). LIPIcs, vol. 168, pp. 43:1–43:19. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2020). https://doi.org/10.4230/LIPICS.ICALP.2020.43

  26. Eppstein, D.: Separating thickness from geometric thickness. In: Kobourov, S.G., Goodrich, M.T. (eds.) Graph Drawing, 10th International Symposium, GD 2002, Irvine, CA, USA, August 26-28, 2002, Revised Papers. Lecture Notes in Computer Science, vol. 2528, pp. 150–161. Springer (2002). https://doi.org/10.1007/3-540-36151-0_15

  27. Fellows, M.R., Lokshtanov, D., Misra, N., Rosamond, F.A., Saurabh, S.: Graph layout problems parameterized by vertex cover. In: Hong, S., Nagamochi, H., Fukunaga, T. (eds.) Algorithms and Computation, 19th International Symposium, ISAAC 2008, Gold Coast, Australia, December 15–17, 2008. Proceedings. Lecture Notes in Computer Science, vol. 5369, pp. 294–305. Springer (2008). https://doi.org/10.1007/978-3-540-92182-0_28

  28. Fomin, F.V., Liedloff, M., Montealegre, P., Todinca, I.: Algorithms parameterized by vertex cover and modular width, through potential maximal cliques. Algorithmica 80(4), 1146–1169 (2018). https://doi.org/10.1007/S00453-017-0297-1

    Article  MathSciNet  MATH  Google Scholar 

  29. Förster, H., Kindermann, P., Miltzow, T., Parada, I., Terziadis, S., Vogtenhuber, B.: Geometric thickness of multigraphs is \(\exists \mathbb{R}\)-complete. In: Soto, J.A., Wiese, A. (eds.) LATIN 2024: Theoretical Informatics - 16th Latin American Symposium, Puerto Varas, Chile, March 18-22, 2024, Proceedings, Part I. Lecture Notes in Computer Science, vol. 14578, pp. 336–349. Springer (2024). https://doi.org/10.1007/978-3-031-55598-5_22

  30. Ganian, R., Hamm, T., Klute, F., Parada, I., Vogtenhuber, B.: Crossing-optimal extension of simple drawings. In: Bansal, N., Merelli, E., Worrell, J. (eds.) 48th International Colloquium on Automata, Languages, and Programming, ICALP 2021, July 12-16, 2021, Glasgow, Scotland (Virtual Conference). LIPIcs, vol. 198, pp. 72:1–72:17. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2021). https://doi.org/10.4230/LIPICS.ICALP.2021.72

  31. Ganian, R., Montecchiani, F., Nöllenburg, M., Zehavi, M.: Parameterized complexity in graph drawing (dagstuhl seminar 21293). Dagstuhl Rep. 11(6), 82–123 (2021). https://doi.org/10.4230/DAGREP.11.6.82

    Article  MATH  Google Scholar 

  32. Goodman, J.E., O’Rourke, J., Tóth, C.D. (eds.): Handbook of discrete and computational geometry. Discrete Mathematics and its Applications (Boca Raton), CRC Press, Boca Raton, FL, third edn. (2018)

    Google Scholar 

  33. Grigoryev, D.Y., Vorobjov, N.N., Jr.: Counting connected components of a semialgebraic set in subexponential time. Comput. Complex. 2, 133–186 (1992). https://doi.org/10.1007/BF01202001

    Article  MathSciNet  MATH  Google Scholar 

  34. Jain, R., Ricci, M., Rollin, J., Schulz, A.: On the geometric thickness of 2-degenerate graphs. In: Chambers, E.W., Gudmundsson, J. (eds.) 39th International Symposium on Computational Geometry, SoCG 2023, June 12-15, 2023, Dallas, Texas, USA. LIPIcs, vol. 258, pp. 44:1–44:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023). https://doi.org/10.4230/LIPICS.SOCG.2023.44

  35. Mansfield, A.: Determining the thickness of graphs is np-hard. In: Mathematical Proceedings of the Cambridge Philosophical Society, vol. 93, pp. 9–23. Cambridge University Press (1983)

    Google Scholar 

  36. Mutzel, P., Odenthal, T., Scharbrodt, M.: The thickness of graphs: a survey. Graphs Comb. 14(1), 59–73 (1998). https://doi.org/10.1007/PL00007219

    Article  MathSciNet  MATH  Google Scholar 

  37. Nešetřil, J., de Mendez, P.O.: Sparsity. Algorithms and Combinatorics 28, xxiv+–457 (2012). https://doi.org/10.1007/978-3-642-27875-4

  38. Nesetril, J., de Mendez, P.O.: Sparsity - Graphs, Structures, and Algorithms, Algorithms and Combinatorics, vol. 28. Springer (2012). https://doi.org/10.1007/978-3-642-27875-4

  39. Robertson, N., Seymour, P.D.: Graph minors. i. excluding a forest. J. Comb. Theory, Ser. B 35(1), 39–61 (1983). https://doi.org/10.1016/0095-8956(83)90079-5

  40. Robertson, N., Seymour, P.D.: Graph minors. II. algorithmic aspects of tree-width. J. Algorithms 7(3), 309–322 (1986). https://doi.org/10.1016/0196-6774(86)90023-4

  41. Schaefer, M.: Complexity of some geometric and topological problems. In: Eppstein, D., Gansner, E.R. (eds.) Graph Drawing (GD’09). LNCS, vol. 5849, pp. 334–344. Springer (2009). https://doi.org/10.1007/978-3-642-11805-0_32

  42. Toth, C.D., O’Rourke, J., Goodman, J.E.: Handbook of discrete and computational geometry. CRC Press (2017)

    Google Scholar 

  43. Tutte, W.T.: The thickness of a graph. Indag. Math. 25, 561–577 (1963)

    MathSciNet  MATH  Google Scholar 

  44. Zehavi, M.: Parameterized analysis and crossing minimization problems. Comput. Sci. Rev. 45, 100490 (2022). https://doi.org/10.1016/J.COSREV.2022.100490

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. TU Wien, Vienna, Austria

    Thomas Depian, Simon Dominik Fink, Alexander Firbas, Robert Ganian & Martin Nöllenburg

Authors
  1. Thomas Depian
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Simon Dominik Fink
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Alexander Firbas
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Robert Ganian
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Martin Nöllenburg
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexander Firbas .

Editor information

Editors and Affiliations

  1. Comenius University in Bratislava, Bratislava, Slovakia

    Rastislav Královič

  2. Czech Academy of Sciences, Prague, Czech Republic

    Věra Kůrková

Rights and permissions

Reprints and permissions

Copyright information

© 2025 The Author(s), under exclusive license to Springer Nature Switzerland AG

About this paper

Check for updates. Verify currency and authenticity via CrossMark

Cite this paper

Depian, T., Fink, S.D., Firbas, A., Ganian, R., Nöllenburg, M. (2025). Pathways to Tractability for Geometric Thickness. In: Královič, R., Kůrková, V. (eds) SOFSEM 2025: Theory and Practice of Computer Science. SOFSEM 2025. Lecture Notes in Computer Science, vol 15538. Springer, Cham. https://doi.org/10.1007/978-3-031-82670-2_16

Download citation

  • DOI: https://doi.org/10.1007/978-3-031-82670-2_16

  • Published:

  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-031-82669-6

  • Online ISBN: 978-3-031-82670-2

  • eBook Packages: Computer ScienceComputer Science (R0)

Publish with us

Policies and ethics