Document Open Access Logo

An SPQR-Tree-Like Embedding Representation for Level Planarity

Authors Guido Brückner , Ignaz Rutter

PDF

File

Thumbnail PDF
PDF
LIPIcs.ISAAC.2020.8.pdf
  • Filesize: 0.68 MB
  • 15 pages

Document Identifiers

Related Versions

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Data structures design and analysis
  • Theory of computation → Graph algorithms analysis
Keywords
  • SPQR-tree
  • Level planarity
  • Partial drawings
  • Simultaneous drawings

Metrics

  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    0
    PDF Downloads
    0
    Metadata Views

Abstract

An SPQR-tree is a data structure that efficiently represents all planar embeddings of a biconnected planar graph. It is a key tool in a number of constrained planarity testing algorithms, which seek a planar embedding of a graph subject to some given set of constraints.
We develop an SPQR-tree-like data structure that represents all level-planar embeddings of a biconnected level graph with a single source, called the LP-tree, and give a simple algorithm to compute it in linear time. Moreover, we show that LP-trees can be used to adapt three constrained planarity algorithms to the level-planar case by using them as a drop-in replacement for SPQR-trees.

Cite As Get BibTex

Guido Brückner and Ignaz Rutter. An SPQR-Tree-Like Embedding Representation for Level Planarity. In 31st International Symposium on Algorithms and Computation (ISAAC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 181, pp. 8:1-8:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ISAAC.2020.8

Author Details

Guido Brückner
  • Karlsruhe Institute of Technology (KIT), Germany
Ignaz Rutter
  • Universität Passau, Germany

Funding

This work was partially supported by DFG grant Ru 1903/3-1.

References

  1. Patrizio Angelini and Michael A. Bekos. Hierarchical partial planarity. Algorithmica, 81(6):2196-2221, June 2019. URL: https://doi.org/10.1007/s00453-018-0530-6.
  2. Patrizio Angelini, Thomas Bläsius, and Ignaz Rutter. Testing mutual duality of planar graphs. International Journal of Computational Geometry & Applications, 24(4):325-346, 2014. URL: https://doi.org/10.1142/S0218195914600103.
  3. Patrizio Angelini, Steven Chaplick, Sabine Cornelsen, Giordano Da Lozzo, Giuseppe Di Battista, Peter Eades, Philipp Kindermann, Jan Kratochvíl, Fabian Lipp, and Ignaz Rutter. Simultaneous orthogonal planarity. In Yifan Hu and Martin Nöllenburg, editors, Graph Drawing and Network Visualization, pages 532-545. Springer, 2016. URL: https://doi.org/10.1007/978-3-319-50106-2_41.
  4. Patrizio Angelini, Giuseppe Di Battista, Fabrizio Frati, Vít Jelínek, Jan Kratochvíl, Maurizio Patrignani, and Ignaz Rutter. Testing planarity of partially embedded graphs. ACM Transactions on Algorithms, 11(4):32:1-32:42, 2015. URL: https://doi.org/10.1145/2629341.
  5. Patrizio Angelini, Giuseppe Di Battista, Fabrizio Frati, Maurizio Patrignani, and Ignaz Rutter. Testing the simultaneous embeddability of two graphs whose intersection is a biconnected or a connected graph. Journal of Discrete Algorithms, 14:150-172, 2012. URL: https://doi.org/10.1016/j.jda.2011.12.015.
  6. Patrizio Angelini, Giuseppe Di Battista, and Maurizio Patrignani. Finding a minimum-depth embedding of a planar graph in O(n⁴) time. Algorithmica, 60(4):890-937, August 2011. URL: https://doi.org/10.1007/s00453-009-9380-6.
  7. Daniel Bienstock and Clyde L. Monma. On the complexity of embedding planar graphs to minimize certain distance measures. Algorithmica, 5(1):93-109, June 1990. URL: https://doi.org/10.1007/BF01840379.
  8. Thomas Bläsius, Stephen G. Kobourov, and Ignaz Rutter. Simultaneous embedding of planar graphs. In Handbook on Graph Drawing and Visualization, pages 349-381. Chapman and Hall/CRC, 2013. Google Scholar
  9. Thomas Bläsius, Ignaz Rutter, and Dorothea Wagner. Optimal orthogonal graph drawing with convex bend costs. ACM Transactions on Algorithms, 12(3), June 2016. URL: https://doi.org/10.1145/2838736.
  10. Guido Brückner, , and Ignaz Rutter. An SPQR-tree-like embedding representation for level planarity, 2020. URL: http://arxiv.org/abs/2009.12309.
  11. Guido Brückner, Markus Himmel, and Ignaz Rutter. An SPQR-tree-like embedding representation for upward planarity. In Daniel Archambault and Csaba D. Tóth, editors, Graph Drawing and Network Visualization, pages 517-531. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-35802-0_39.
  12. Guido Brückner and Ignaz Rutter. Partial and constrained level planarity. In Philip N. Klein, editor, Proceedings of the 28th Annual ACM-SIAM Symposium on Discrete Algorithms, pages 2000-2011. SIAM, 2017. URL: https://doi.org/10.1137/1.9781611974782.130.
  13. Markus Chimani, Carsten Gutwenger, Petra Mutzel, and Christian Wolf. Inserting a vertex into a planar graph. In Proceedings of the Twentieth Annual ACM-SIAM Symposium on Discrete Algorithms, pages 375-383. SIAM, 2009. Google Scholar
  14. Markus Chimani and Petr Hlinený. Inserting multiple edges into a planar graph. In Sándor Fekete and Anna Lubiw, editors, 32nd International Symposium on Computational Geometry, volume 51, pages 30:1-30:15, 2016. URL: https://doi.org/10.4230/LIPIcs.SoCG.2016.30.
  15. Giuseppe Di Battista and Enrico Nardelli. Hierarchies and planarity theory. IEEE Transactions on Systems, Man, and Cybernetics, 18(6):1035-1046, 1988. URL: https://doi.org/10.1109/21.23105.
  16. Giuseppe Di Battista and Roberto Tamassia. Incremental planarity testing. In Proceedings of the 30th Annual Symposium on Foundations of Computer Science, pages 436-441, October 1989. URL: https://doi.org/10.1109/SFCS.1989.63515.
  17. Giuseppe Di Battista and Roberto Tamassia. On-line graph algorithms with SPQR-trees. In Michael S. Paterson, editor, Proceedings of the 17th International Colloquium on Automata, Languages and Programming, pages 598-611. Springer Berlin Heidelberg, 1990. URL: https://doi.org/10.1007/BFb0032061.
  18. Giuseppe Di Battista and Roberto Tamassia. On-line maintenance of triconnected components with SPQR-trees. Algorithmica, 15(4):302-318, 1996. URL: https://doi.org/10.1007/BF01961541.
  19. Radoslav Fulek, Michael J. Pelsmajer, Marcus Schaefer, and Daniel Stefankovic. Hanani-Tutte and monotone drawings. In Petr Kolman and Jan Kratochvíl, editors, Proceedings of the 37th International Workshop on Graph-Theoretic Concepts in Computer Science, volume 6986 of Lecture Notes in Computer Science, pages 283-294. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-25870-1_26.
  20. Carsten Gutwenger, Petra Mutzel, and René Weiskircher. Inserting an edge into a planar graph. Algorithmica, 41(4):289-308, 2005. URL: https://doi.org/10.1007/s00453-004-1128-8.
  21. Seok-Hee Hong, Brendan McKay, and Peter Eades. A linear time algorithm for constructing maximally symmetric straight line drawings of triconnected planar graphs. Discrete & Computational Geometry, 36(2):283-311, September 2006. URL: https://doi.org/10.1007/s00454-006-1231-5.
  22. John Edward Hopcroft and Robert Endre Tarjan. Dividing a graph into triconnected components. SIAM Journal on Computing, 2(3):135-158, 1973. URL: https://doi.org/10.1137/0202012.
  23. Michael D. Hutton and Anna Lubiw. Upward planar drawing of single-source acyclic digraphs. SIAM Journal on Computing, 25(2):291-–311, February 1996. URL: https://doi.org/10.1137/S0097539792235906.
  24. Michael Jünger and Sebastian Leipert. Level planar embedding in linear time. Journal of Graph Algorithms and Applications, 6(1):67-113, 2002. URL: https://doi.org/10.7155/jgaa.00045.
  25. Michael Jünger, Sebastian Leipert, and Petra Mutzel. Level planarity testing in linear time. In Sue H. Whitesides, editor, Graph Drawing, pages 224-237. Springer, 1998. Google Scholar
  26. Saunders Mac Lane. A structural characterization of planar combinatorial graphs. Duke Mathematical Journal, 3(3):460-472, 1937. URL: https://doi.org/10.1215/S0012-7094-37-00336-3.
  27. Bert Randerath, Ewald Speckenmeyer, Endre Boros, Peter Hammer, Alex Kogan, Kazuhisa Makino, Bruno Simeone, and Ondrej Cepek. A satisfiability formulation of problems on level graphs. Electronic Notes in Discrete Mathematics, 9:269-277, 2001. Google Scholar
  28. Roberto Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM Journal on Computing, 16(3):421-444, 1987. URL: https://doi.org/10.1137/0216030.
  29. William Thomas Tutte. Connectivity in Graphs. University of Toronto Press, 1966. Google Scholar
Questions / Remarks / Feedback
X

Feedback for Dagstuhl Publishing


Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail
© 2023-2025 Schloss Dagstuhl – LZI GmbH About DROPS Imprint Privacy Contact