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Rectilinear-Upward Planarity Testing of Digraphs

Authors Walter Didimo , Michael Kaufmann , Giuseppe Liotta , Giacomo Ortali , Maurizio Patrignani



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Author Details

Walter Didimo
  • Department of Engineering, University of Perugia, Italy
Michael Kaufmann
  • Department of Computer Science, University of Tübingen, Germany
Giuseppe Liotta
  • Department of Engineering, University of Perugia, Italy
Giacomo Ortali
  • Department of Engineering, University of Perugia, Italy
Maurizio Patrignani
  • Department of Civil, Computer and Aeronautical Engineering, Roma Tre University, Italy

Acknowledgements

We thank Ignaz Rutter for conversations about the problem 1-2-Switch-Flow.

Cite AsGet BibTex

Walter Didimo, Michael Kaufmann, Giuseppe Liotta, Giacomo Ortali, and Maurizio Patrignani. Rectilinear-Upward Planarity Testing of Digraphs. In 34th International Symposium on Algorithms and Computation (ISAAC 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 283, pp. 26:1-26:20, Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ISAAC.2023.26

Abstract

A rectilinear-upward planar drawing of a digraph G is a crossing-free drawing of G where each edge is either a horizontal or a vertical segment, and such that no directed edge points downward. Rectilinear-Upward Planarity Testing is the problem of deciding whether a digraph G admits a rectilinear-upward planar drawing. We show that: (i) Rectilinear-Upward Planarity Testing is NP-complete, even if G is biconnected; (ii) it can be solved in linear time when an upward planar embedding of G is fixed; (iii) the problem is polynomial-time solvable for biconnected digraphs of treewidth at most two, i.e., for digraphs whose underlying undirected graph is a series-parallel graph; (iv) for any biconnected digraph the problem is fixed-parameter tractable when parameterized by the number of sources and sinks in the digraph.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Dynamic programming
  • Theory of computation → Graph algorithms analysis
Keywords
  • Graph drawing
  • orthogonal drawings
  • upward drawings
  • rectilinear planarity
  • upward planarity

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References

  1. Sarmad Abbasi, Patrick Healy, and Aimal Rextin. Improving the running time of embedded upward planarity testing. Inf. Process. Lett., 110(7):274-278, 2010. URL: https://doi.org/10.1016/j.ipl.2010.02.004.
  2. Giuseppe Di Battista and Roberto Tamassia. Algorithms for plane representations of acyclic digraphs. Theor. Comput. Sci., 61:175-198, 1988. URL: https://doi.org/10.1016/0304-3975(88)90123-5.
  3. Paola Bertolazzi, Giuseppe Di Battista, Giuseppe Liotta, and Carlo Mannino. Upward drawings of triconnected digraphs. Algorithmica, 12(6):476-497, 1994. URL: https://doi.org/10.1007/BF01188716.
  4. Paola Bertolazzi, Giuseppe Di Battista, Carlo Mannino, and Roberto Tamassia. Optimal upward planarity testing of single-source digraphs. SIAM J. Comput., 27(1):132-169, 1998. URL: https://doi.org/10.1137/S0097539794279626.
  5. Carla Binucci, Walter Didimo, and Maurizio Patrignani. Upward and quasi-upward planarity testing of embedded mixed graphs. Theor. Comput. Sci., 526:75-89, 2014. URL: https://doi.org/10.1016/j.tcs.2014.01.015.
  6. Franz-Josef Brandenburg, David Eppstein, Michael T. Goodrich, Stephen G. Kobourov, Giuseppe Liotta, and Petra Mutzel. Selected open problems in graph drawing. In Giuseppe Liotta, editor, Graph Drawing, 11th International Symposium, GD 2003, Perugia, Italy, September 21-24, 2003, Proceedings, volume 2912 of Lecture Notes in Computer Science, pages 515-539. Springer, 2003. URL: https://doi.org/10.1007/978-3-540-24595-7_55.
  7. Steven Chaplick, Emilio Di Giacomo, Fabrizio Frati, Robert Ganian, Chrysanthi N. Raftopoulou, and Kirill Simonov. Parameterized algorithms for upward planarity. In Xavier Goaoc and Michael Kerber, editors, 38th International Symposium on Computational Geometry, SoCG 2022, June 7-10, 2022, Berlin, Germany, volume 224 of LIPIcs, pages 26:1-26:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.SoCG.2022.26.
  8. Steven Chaplick, Emilio Di Giacomo, Fabrizio Frati, Robert Ganian, Chrysanthi N. Raftopoulou, and Kirill Simonov. Testing upward planarity of partial 2-trees. In Patrizio Angelini and Reinhard von Hanxleden, editors, Graph Drawing and Network Visualization - 30th International Symposium, GD 2022, Tokyo, Japan, September 13-16, 2022, Proceedings, volume 13764 of Lecture Notes in Computer Science, pages 175-187. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-22203-0_13.
  9. Sabine Cornelsen and Andreas Karrenbauer. Accelerated bend minimization. J. Graph Algorithms Appl., 16(3):635-650, 2012. URL: https://doi.org/10.7155/jgaa.00265.
  10. Giuseppe Di Battista, Peter Eades, Roberto Tamassia, and Ioannis G. Tollis. Graph Drawing: Algorithms for the Visualization of Graphs. Prentice-Hall, 1999. Google Scholar
  11. Giuseppe Di Battista, Giuseppe Liotta, and Francesco Vargiu. Spirality and optimal orthogonal drawings. SIAM J. Comput., 27(6):1764-1811, 1998. URL: https://doi.org/10.1137/S0097539794262847.
  12. Emilio Di Giacomo, Giuseppe Liotta, and Fabrizio Montecchiani. Sketched representations and orthogonal planarity of bounded treewidth graphs. In Daniel Archambault and Csaba D. Tóth, editors, Graph Drawing and Network Visualization - 27th International Symposium, GD 2019, Prague, Czech Republic, September 17-20, 2019, Proceedings, volume 11904 of Lecture Notes in Computer Science, pages 379-392. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-35802-0_29.
  13. Emilio Di Giacomo, Giuseppe Liotta, and Fabrizio Montecchiani. Orthogonal planarity testing of bounded treewidth graphs. J. Comput. Syst. Sci., 125:129-148, 2022. URL: https://doi.org/10.1016/j.jcss.2021.11.004.
  14. Walter Didimo, Emilio Di Giacomo, Giuseppe Liotta, Fabrizio Montecchiani, and Giacomo Ortali. On the parameterized complexity of bend-minimum orthogonal planarity. CoRR, abs/2308.13665, 2023. URL: https://doi.org/10.48550/arXiv.2308.13665.
  15. Walter Didimo, Francesco Giordano, and Giuseppe Liotta. Upward spirality and upward planarity testing. SIAM J. Discret. Math., 23(4):1842-1899, 2009. URL: https://doi.org/10.1137/070696854.
  16. Walter Didimo, Michael Kaufmann, Giuseppe Liotta, and Giacomo Ortali. Rectilinear planarity of partial 2-trees. In Patrizio Angelini and Reinhard von Hanxleden, editors, Graph Drawing and Network Visualization - 30th International Symposium, GD 2022, Tokyo, Japan, September 13-16, 2022, Proceedings, volume 13764 of Lecture Notes in Computer Science, pages 157-172. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-22203-0_12.
  17. Walter Didimo, Michael Kaufmann, Giuseppe Liotta, and Giacomo Ortali. Rectilinear planarity of partial 2-trees. CoRR, abs/2208.12558, 2022. URL: https://doi.org/10.48550/arXiv.2208.12558.
  18. Walter Didimo, Michael Kaufmann, Giuseppe Liotta, and Giacomo Ortali. Computing bend-minimum orthogonal drawings of plane series–parallel graphs in linear time. Algorithmica, 2023. URL: https://doi.org/10.1007/s00453-023-01110-6.
  19. Walter Didimo, Michael Kaufmann, Giuseppe Liotta, and Giacomo Ortali. Rectilinear planarity of partial 2-trees. J. of Graph Algorithms Appl., special issue on GD 2022., 2023. to appear. Google Scholar
  20. Walter Didimo and Giuseppe Liotta. Computing orthogonal drawings in a variable embedding setting. In Kyung-Yong Chwa and Oscar H. Ibarra, editors, Algorithms and Computation, 9th International Symposium, ISAAC '98, Taejon, Korea, December 14-16, 1998, Proceedings, volume 1533 of Lecture Notes in Computer Science, pages 79-88. Springer, 1998. URL: https://doi.org/10.1007/3-540-49381-6_10.
  21. Walter Didimo and Giuseppe Liotta. Graph Visualization and Data Mining, chapter 3, pages 35-63. John Wiley & Sons, Ltd, 2006. URL: https://doi.org/10.1002/9780470073049.ch3.
  22. Walter Didimo, Giuseppe Liotta, Giacomo Ortali, and Maurizio Patrignani. Optimal orthogonal drawings of planar 3-graphs in linear time. In Shuchi Chawla, editor, Proceedings of the 2020 ACM-SIAM Symposium on Discrete Algorithms, SODA 2020, Salt Lake City, UT, USA, January 5-8, 2020, pages 806-825. SIAM, 2020. URL: https://doi.org/10.1137/1.9781611975994.49.
  23. Walter Didimo, Giuseppe Liotta, and Maurizio Patrignani. HV-planarity: Algorithms and complexity. J. Comput. Syst. Sci., 99:72-90, 2019. URL: https://doi.org/10.1016/j.jcss.2018.08.003.
  24. Shimon Even and Robert Endre Tarjan. Corrigendum: Computing an st-numbering. TCS 2(1976):339-344. Theor. Comput. Sci., 4(1):123, 1977. Google Scholar
  25. Uli Fossmeier and Michael Kaufmann. An approach to bend-minimal upward drawing. In Graph Drawing, 1th International Symposium, GD 1993, Paris, France, pages 27-29, 1993. Google Scholar
  26. Ulrich Fößmeier and Michael Kaufmann. On bend-minimum orthogonal upward drawing of directed planar graphs. In Roberto Tamassia and Ioannis G. Tollis, editors, Graph Drawing, DIMACS International Workshop, GD '94, Princeton, New Jersey, USA, October 10-12, 1994, Proceedings, volume 894 of Lecture Notes in Computer Science, pages 52-63. Springer, 1994. URL: https://doi.org/10.1007/3-540-58950-3_356.
  27. Fabrizio Frati. Planar rectilinear drawings of outerplanar graphs in linear time. Comput. Geom., 103:101854, 2022. URL: https://doi.org/10.1016/j.comgeo.2021.101854.
  28. Ashim Garg and Roberto Tamassia. A new minimum cost flow algorithm with applications to graph drawing. In Stephen C. North, editor, Graph Drawing, Symposium on Graph Drawing, GD '96, Berkeley, California, USA, September 18-20, Proceedings, volume 1190 of Lecture Notes in Computer Science, pages 201-216. Springer, 1996. URL: https://doi.org/10.1007/3-540-62495-3_49.
  29. Ashim Garg and Roberto Tamassia. On the computational complexity of upward and rectilinear planarity testing. SIAM J. Comput., 31(2):601-625, 2001. URL: https://doi.org/10.1137/S0097539794277123.
  30. Carsten Gutwenger and Petra Mutzel. A linear time implementation of spqr-trees. In Joe Marks, editor, Graph Drawing, 8th International Symposium, GD 2000, Colonial Williamsburg, VA, USA, September 20-23, 2000, Proceedings, volume 1984 of Lecture Notes in Computer Science, pages 77-90. Springer, 2000. URL: https://doi.org/10.1007/3-540-44541-2_8.
  31. Md. Manzurul Hasan and Md. Saidur Rahman. No-bend orthogonal drawings and no-bend orthogonally convex drawings of planar graphs (extended abstract). In Ding-Zhu Du, Zhenhua Duan, and Cong Tian, editors, Computing and Combinatorics - 25th International Conference, COCOON 2019, Xi'an, China, July 29-31, 2019, Proceedings, volume 11653 of Lecture Notes in Computer Science, pages 254-265. Springer, 2019. URL: https://doi.org/10.1007/978-3-030-26176-4_21.
  32. Patrick Healy and Karol Lynch. Two fixed-parameter tractable algorithms for testing upward planarity. Int. J. Found. Comput. Sci., 17(5):1095-1114, 2006. URL: https://doi.org/10.1142/S0129054106004285.
  33. John E. Hopcroft and Robert Endre Tarjan. Dividing a graph into triconnected components. SIAM J. Comput., 2(3):135-158, 1973. URL: https://doi.org/10.1137/0202012.
  34. Michael D. Hutton and Anna Lubiw. Upward planning of single-source acyclic digraphs. SIAM J. Comput., 25(2):291-311, 1996. URL: https://doi.org/10.1137/S0097539792235906.
  35. Bart M. P. Jansen, Liana Khazaliya, Philipp Kindermann, Giuseppe Liotta, Fabrizio Montecchiani, and Kirill Simonov. Upward and orthogonal planarity are W[1]-hard parameterized by treewidth. CoRR, abs/2309.01264, 2023. URL: https://doi.org/10.48550/arXiv.2309.01264.
  36. Michael Jünger and Petra Mutzel, editors. Graph Drawing Software. Springer, 2004. URL: https://doi.org/10.1007/978-3-642-18638-7.
  37. M. R. Krom. The decision problem for a class of first-order formulas in which all disjunctions are binary. Mathematical Logic Quarterly, 13(1-2):15-20, 1967. URL: https://doi.org/10.1002/malq.19670130104.
  38. Takao Nishizeki and Md. Saidur Rahman. Planar Graph Drawing, volume 12 of Lecture Notes Series on Computing. World Scientific, 2004. URL: https://doi.org/10.1142/5648.
  39. Maurizio Patrignani. Planarity testing and embedding. In Roberto Tamassia, editor, Handbook on Graph Drawing and Visualization, pages 1-42. Chapman and Hall/CRC, 2013. Google Scholar
  40. Md. Saidur Rahman, Noritsugu Egi, and Takao Nishizeki. No-bend orthogonal drawings of subdivisions of planar triconnected cubic graphs. IEICE Trans. Inf. Syst., 88-D(1):23-30, 2005. URL: http://search.ieice.org/bin/summary.php?id=e88-d_1_23&category=D&year=2005&lang=E&abst=.
  41. Md. Saidur Rahman, Takao Nishizeki, and Mahmuda Naznin. Orthogonal drawings of plane graphs without bends. J. Graph Algorithms Appl., 7(4):335-362, 2003. URL: https://doi.org/10.7155/jgaa.00074.
  42. Roberto Tamassia. On embedding a graph in the grid with the minimum number of bends. SIAM J. Comput., 16(3):421-444, 1987. URL: https://doi.org/10.1137/0216030.
  43. Roberto Tamassia and Giuseppe Liotta. Graph drawing. In Jacob E. Goodman and Joseph O'Rourke, editors, Handbook of Discrete and Computational Geometry, Second Edition, pages 1163-1185. Chapman and Hall/CRC, 2004. URL: https://doi.org/10.1201/9781420035315.ch52.
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