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Monotone Arc Diagrams with Few Biarcs

Authors Steven Chaplick , Henry Förster , Michael Hoffmann , Michael Kaufmann

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Human-centered computing → Graph drawings
Keywords
  • planar graph
  • topological book embedding
  • monotone drawing
  • linear layout

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Abstract

We show that every planar graph has a monotone topological 2-page book embedding where at most (4n-10)/5 (of potentially 3n-6) edges cross the spine, and every edge crosses the spine at most once; such an edge is called a biarc. We can also guarantee that all edges that cross the spine cross it in the same direction (e.g., from bottom to top). For planar 3-trees we can further improve the bound to (3n-9)/4, and for so-called Kleetopes we obtain a bound of at most (n-8)/3 edges that cross the spine. The bound for Kleetopes is tight, even if the drawing is not required to be monotone. A Kleetope is a plane triangulation that is derived from another plane triangulation T by inserting a new vertex v_f into each face f of T and then connecting v_f to the three vertices of f.

Cite As Get BibTex

Steven Chaplick, Henry Förster, Michael Hoffmann, and Michael Kaufmann. Monotone Arc Diagrams with Few Biarcs. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 11:1-11:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024) https://doi.org/10.4230/LIPIcs.GD.2024.11

Author Details

Steven Chaplick
  • Maastricht University, The Netherlands
Henry Förster
  • Universität Tübingen, Germany
Michael Hoffmann
  • Department of Computer Science, ETH Zürich, Switzerland
Michael Kaufmann
  • Universität Tübingen, Germany

Funding

  • Chaplick, Steven: supported by DFG grant WO 758/11-1.
  • Hoffmann, Michael: supported by the Swiss National Science Foundation within the collaborative D-A-CH project Arrangements and Drawings as SNSF project 200021E-171681.

Acknowledgements

This work started at the workshop on Graph and Network Visualization (GNV 2017) in Heiligkreuztal, Germany. Preliminary results were presented at the 36th European Workshop on Computational Geometry (EuroCG 2020). We thank Stefan Felsner and Stephen Kobourov for useful discussions.

References

  1. Frank Bernhart and Paul C. Kainen. The book thickness of a graph. J. Combin. Theory Ser. B, 27:320-331, 1979. URL: https://doi.org/10.1016/0095-8956(79)90021-2.
  2. Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein. Arc diagrams, flip distances, and Hamiltonian triangulations. In Proc. 32nd Int. Sympos. Theoret. Aspects Comput. Sci. (STACS 2015), volume 30 of LIPIcs, pages 197-210, 2015. URL: https://doi.org/10.4230/LIPIcs.STACS.2015.197.
  3. Jean Cardinal, Michael Hoffmann, Vincent Kusters, Csaba D. Tóth, and Manuel Wettstein. Arc diagrams, flip distances, and Hamiltonian triangulations. Comput. Geom. Theory Appl., 68:206-225, 2018. URL: https://doi.org/10.1016/j.comgeo.2017.06.001.
  4. Marek Chrobak and Thomas H. Payne. A linear-time algorithm for drawing a planar graph on a grid. Inform. Process. Lett., 54:241-246, 1995. URL: https://doi.org/10.1016/0020-0190(95)00020-D.
  5. Hubert de Fraysseix, János Pach, and Richard Pollack. Small sets supporting Fary embeddings of planar graphs. In Proc. 20th Annu. ACM Sympos. Theory Comput. (STOC 1988), pages 426-433, 1988. URL: https://doi.org/10.1145/62212.62254.
  6. Hubert de Fraysseix, János Pach, and Richard Pollack. How to draw a planar graph on a grid. Combinatorica, 10(1):41-51, 1990. URL: https://doi.org/10.1007/BF02122694.
  7. Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, and Stephen K. Wismath. Curve-constrained drawings of planar graphs. Comput. Geom. Theory Appl., 30(1):1-23, 2005. URL: https://doi.org/10.1016/j.comgeo.2004.04.002.
  8. Hazel Everett, Sylvain Lazard, Giuseppe Liotta, and Stephen K. Wismath. Universal sets of n points for 1-bend drawings of planar graphs with n vertices. In Proc. 15th Int. Sympos. Graph Drawing (GD 2007), volume 4875 of LNCS, pages 345-351, 2007. URL: https://doi.org/10.1007/978-3-540-77537-9_34.
  9. Hazel Everett, Sylvain Lazard, Giuseppe Liotta, and Stephen K. Wismath. Universal sets of n points for one-bend drawings of planar graphs with n vertices. Discrete Comput. Geom., 43(2):272-288, 2010. URL: https://doi.org/10.1007/s00454-009-9149-3.
  10. Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, and Stephen K. Wismath. Drawing planar graphs on a curve. In Proc. 13th Int. Workshop Graph-Theoret. Concepts Comput. Sci. (WG 2003), volume 2880 of LNCS, pages 192-204, 2003. URL: https://doi.org/10.1007/978-3-540-39890-5_17.
  11. Francesco Giordano, Giuseppe Liotta, Tamara Mchedlidze, Antonios Symvonis, and Sue Whitesides. Computing upward topological book embeddings of upward planar digraphs. J. Discrete Algorithms, 30:45-69, 2015. URL: https://doi.org/10.1016/j.jda.2014.11.006.
  12. Michael Kaufmann and Roland Wiese. Embedding vertices at points: Few bends suffice for planar graphs. In Proc. 6th Int. Sympos. Graph Drawing (GD 1998), volume 1547 of LNCS, pages 462-463, 1998. URL: https://doi.org/10.1007/3-540-37623-2_47.
  13. Michael Kaufmann and Roland Wiese. Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl., 6(1):115-129, 2002. URL: https://doi.org/10.7155/jgaa.00046.
  14. Maarten Löffler and Csaba D. Tóth. Linear-size universal point sets for one-bend drawings. In Proc. 23rd Int. Sympos. Graph Drawing Network Visualization (GD 2015), volume 9411 of LNCS, pages 423-429, 2015. URL: https://doi.org/10.1007/978-3-319-27261-0_35.
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