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The Price of Upwardness

Authors Patrizio Angelini , Therese Biedl , Markus Chimani , Sabine Cornelsen , Giordano Da Lozzo , Seok-Hee Hong , Giuseppe Liotta , Maurizio Patrignani , Sergey Pupyrev , Ignaz Rutter , Alexander Wolff

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Author Details

Patrizio Angelini
  • John Cabot University, Rome, Italy
Therese Biedl
  • University of Waterloo, Canada
Markus Chimani
  • University of Osnabrück, Germany
Sabine Cornelsen
  • University of Konstanz, Germany
Giordano Da Lozzo
  • Roma Tre University, Italy
Seok-Hee Hong
  • University of Sydney, Australia
Giuseppe Liotta
  • University of Perugia, Italy
Maurizio Patrignani
  • Roma Tre University, Italy
Sergey Pupyrev
  • Menlo Park, CA, USA
Ignaz Rutter
  • University of Passau, Germany
Alexander Wolff
  • University of Würzburg, Germany

Funding

Research by {Da Lozzo}, {Liotta}, and {Patrignani} was supported, in part, by MUR of Italy (PRIN Project no. 2022ME9Z78 - NextGRAAL and PRIN Project no. 2022TS4Y3N - EXPAND). Research by {Liotta} was supported in part by MUR PON Proj. ARS01_00540. Research by Rutter was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) - 541433306. Research by Biedl was supported by NSERC FRN RGPIN-2020-03958.

Acknowledgements

This work was initiated at the Dagstuhl Seminar "Beyond-Planar Graphs: Models, Structures and Geometric Representations" (No. 24062), February 2024.

Cite AsGet BibTex

Patrizio Angelini, Therese Biedl, Markus Chimani, Sabine Cornelsen, Giordano Da Lozzo, Seok-Hee Hong, Giuseppe Liotta, Maurizio Patrignani, Sergey Pupyrev, Ignaz Rutter, and Alexander Wolff. The Price of Upwardness. In 32nd International Symposium on Graph Drawing and Network Visualization (GD 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 320, pp. 13:1-13:20, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.GD.2024.13

Abstract

Not every directed acyclic graph (DAG) whose underlying undirected graph is planar admits an upward planar drawing. We are interested in pushing the notion of upward drawings beyond planarity by considering upward k-planar drawings of DAGs in which the edges are monotonically increasing in a common direction and every edge is crossed at most k times for some integer k ≥ 1. We show that the number of crossings per edge in a monotone drawing is in general unbounded for the class of bipartite outerplanar, cubic, or bounded pathwidth DAGs. However, it is at most two for outerpaths and it is at most quadratic in the bandwidth in general. From the computational point of view, we prove that upward-k-planarity testing is NP-complete already for k = 1 and even for restricted instances for which upward planarity testing is polynomial. On the positive side, we can decide in linear time whether a single-source DAG admits an upward 1-planar drawing in which all vertices are incident to the outer face.

Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Graph theory
Keywords
  • upward drawings
  • beyond planarity
  • upward k-planarity
  • upward outer-1-planarity

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References

  1. Patrizio Angelini, Therese Biedl, Markus Chimani, Sabine Cornelsen, Giordano Da Lozzo, Seok-Hee Hong, Giuseppe Liotta, Maurizio Patrignani, Sergey Pupyrev, Ignaz Rutter, and Alexander Wolff. The price of upwardness. arXiv 2409.01475. URL: http://arxiv.org/abs/2409.01475.
  2. Patrizio Angelini, Luca Cittadini, Giuseppe Di Battista, Walter Didimo, Fabrizio Frati, Michael Kaufmann, and Antonios Symvonis. On the perspectives opened by right angle crossing drawings. In David Eppstein and Emden R. Gansner, editors, Proc. 17th Int. Symp. Graph Drawing (GD), volume 5849 of LNCS, pages 21-32. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-11805-0_5.
  3. Patrizio Angelini, Luca Cittadini, Walter Didimo, Fabrizio Frati, Giuseppe Di Battista, Michael Kaufmann, and Antonios Symvonis. On the perspectives opened by right angle crossing drawings. J. Graph Algorithms Appl., 15(1):53-78, 2011. URL: https://doi.org/10.7155/JGAA.00217.
  4. Christopher Auer, Christian Bachmaier, Franz J. Brandenburg, Andreas Gleißner, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber. Recognizing outer 1-planar graphs in linear time. In Stephen K. Wismath and Alexander Wolff, editors, Proc. 21st Int. Symp. Graph Drawing (GD), volume 8242 of LNCS, pages 107-118. Springer, 2013. URL: https://doi.org/10.1007/978-3-319-03841-4_10.
  5. 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.
  6. Christopher Auer, Christian Bachmaier, Franz J. Brandenburg, Andreas Gleißner, Kathrin Hanauer, Daniel Neuwirth, and Josef Reislhuber. Correction to: Outer 1-planar graphs. Algorithmica, 83(11):3534-3535, 2021. URL: https://doi.org/10.1007/S00453-021-00874-Z.
  7. Michael A. Bekos, Giordano Da Lozzo, Fabrizio Frati, Martin Gronemann, Tamara Mchedlidze, and Chrysanthi N. Raftopoulou. Recognizing DAGs with page-number 2 is NP-complete. In Patrizio Angelini and Reinhard von Hanxleden, editors, Proc. 30th Int. Symp. Graph Drawing & Netw. Vis. (GD), volume 13764 of LNCS, pages 361-370. Springer, 2022. URL: https://doi.org/10.1007/978-3-031-22203-0_26.
  8. Michael A. Bekos, Giordano Da Lozzo, Fabrizio Frati, Martin Gronemann, Tamara Mchedlidze, and Chrysanthi N. Raftopoulou. Recognizing DAGs with page-number 2 is NP-complete. Theor. Comput. Sci., 946:113689, 2023. URL: https://doi.org/10.1016/J.TCS.2023.113689.
  9. 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.
  10. Sujoy Bhore, Giordano Da Lozzo, Fabrizio Montecchiani, and Martin Nöllenburg. On the upward book thickness problem: Combinatorial and complexity results. In Helen C. Purchase and Ignaz Rutter, editors, Proc. 29th Int. Symp. Graph Drawing & Netw. Vis. (GD), volume 12868 of LNCS, pages 242-256. Springer, 2021. URL: https://doi.org/10.1007/978-3-030-92931-2_18.
  11. Sujoy Bhore, Giordano Da Lozzo, Fabrizio Montecchiani, and Martin Nöllenburg. On the upward book thickness problem: Combinatorial and complexity results. Eur. J. Comb., 110:103662, 2023. URL: https://doi.org/10.1016/J.EJC.2022.103662.
  12. Carla Binucci, Giordano Da Lozzo, Emilio Di Giacomo, Walter Didimo, Tamara Mchedlidze, and Maurizio Patrignani. Upward book embeddability of st-graphs: Complexity and algorithms. Algorithmica, 85(12):3521-3571, 2023. URL: https://doi.org/10.1007/S00453-023-01142-Y.
  13. Julia Böttcher, Klaas P. Pruessmann, Anusch Taraz, and Andreas Würfl. Bandwidth, expansion, treewidth, separators and universality for bounded-degree graphs. Europ. J. Combin., 31(5):1217-1227, 2010. URL: https://doi.org/10.1016/J.EJC.2009.10.010.
  14. 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, Proc. 38th Int. Symp. Comput. Geom. (SoCG), 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.
  15. 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, Proc. 30th Int. Symp. Graph Drawing & Netw. Vis. (GD), volume 13764 of LNCS, pages 175-187, 2022. URL: https://doi.org/10.1007/978-3-031-22203-0_13.
  16. Marek Cygan, Fedor V. Fomin, Łukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. URL: https://doi.org/10.1007/978-3-319-21275-3.
  17. Giuseppe Di Battista and Roberto Tamassia. On-line planarity testing. SIAM J. Comput., 25:956-997, 1996. URL: https://doi.org/10.1137/S0097539794280736.
  18. Walter Didimo, Francesco Giordano, and Giuseppe Liotta. Upward spirality and upward planarity testing. SIAM J. Discrete Math., 23(4):1842-1899, 2010. URL: https://doi.org/10.1137/070696854.
  19. Walter Didimo, Giuseppe Liotta, and Fabrizio Montecchiani. A survey on graph drawing beyond planarity. ACM Comput. Surv., 52(1):4:1-4:37, 2019. URL: https://doi.org/10.1145/3301281.
  20. Vida Dujmović, Ken-ichi Kawarabayashi, Bojan Mohar, and David R. Wood. Improved upper bounds on the crossing number. In Monique Teillaud, editor, Proc. 24th ACM Symp. Comput. Geom. (SoCG), pages 375-384, 2008. URL: https://doi.org/10.1145/1377676.1377739.
  21. Fedor V. Fomin and Petr A. Golovach. Interval degree and bandwidth of a graph. Discrete Appl. Math., 129(2-3):345-359, 2003. URL: https://doi.org/10.1016/S0166-218X(02)00574-7.
  22. Fabrizio Frati, Radoslav Fulek, and Andres J. Ruiz-Vargas. On the page number of upward planar directed acyclic graphs. In Marc J. van Kreveld and Bettina Speckmann, editors, Proc. 19th Int. Symp. Graph Drawing (GD), volume 7034 of LNCS, pages 391-402. Springer, 2011. URL: https://doi.org/10.1007/978-3-642-25878-7_37.
  23. Fabrizio Frati, Radoslav Fulek, and Andres J. Ruiz-Vargas. On the page number of upward planar directed acyclic graphs. J. Graph Algorithms Appl., 17(3):221-244, 2013. URL: https://doi.org/10.7155/JGAA.00292.
  24. Radoslav Fulek, Michael J. Pelsmajer, Marcus Schaefer, and Daniel Štefankovič. Hanani-Tutte, monotone drawings, and level-planarity. In János Pach, editor, Thirty Essays on Geometric Graph Theory, pages 263-287. Springer, 2013. URL: https://doi.org/10.1007/978-1-4614-0110-0_14.
  25. Michael R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  26. Ashim Garg and Roberto Tamassia. Upward planarity testing. Order, 12:109-133, 1995. URL: https://doi.org/10.1007/BF01108622.
  27. 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.
  28. 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.
  29. Lenwood S. Heath, Sriram V. Pemmaraju, and Ann N. Trenk. Stack and queue layouts of directed acyclic graphs: Part I. SIAM J. Comput., 28(4):1510-1539, 1999. URL: https://doi.org/10.1137/S0097539795280287.
  30. Seok-Hee Hong and Takeshi Tokuyama, editors. Beyond Planar Graphs, Communications of NII Shonan Meetings. Springer, 2020. URL: https://doi.org/10.1007/978-981-15-6533-5.
  31. Paul Jungeblut, Laura Merker, and Torsten Ueckerdt. Directed acyclic outerplanar graphs have constant stack number. In Proc. 64th IEEE Ann. Symp. Foundat. Comput. Sci. (FOCS), pages 1937-1952, 2023. URL: https://doi.org/10.1109/FOCS57990.2023.00118.
  32. Stephen G. Kobourov, Giuseppe Liotta, and Fabrizio Montecchiani. An annotated bibliography on 1-planarity. Comput. Sci. Rev., 25:49-67, 2017. URL: https://doi.org/10.1016/J.COSREV.2017.06.002.
  33. Tamara Mchedlidze and Antonios Symvonis. Crossing-optimal acyclic Hamiltonian path completion and its application to upward topological book embeddings. In S. Das and Ryuhei Uehara, editors, Proc. Workshop Algorithms & Comput. (WALCOM), volume 5431 of LNCS, pages 250-261. Springer, 2009. URL: https://doi.org/10.1007/978-3-642-00202-1_22.
  34. Martin Nöllenburg and Sergey Pupyrev. On families of planar DAGs with constant stack number. In Michael A. Bekos and Markus Chimani, editors, Proc. Int. Symp. Graph Drawing & Netw. Vis. (GD), volume 14465 of LNCS, pages 135-151. Springer, 2023. URL: https://doi.org/10.1007/978-3-031-49272-3_10.
  35. Achilleas Papakostas. Upward planarity testing of outerplanar DAGs. In Roberto Tamassia and Ioanni G. Tollis, editors, Proc. Int. Sympos. Graph Drawing (GD), volume 894 of LNCS, pages 298-306. Springer, 1994. URL: https://doi.org/10.1007/3-540-58950-3_385.
  36. Marcus Schaefer. The graph crossing number and its variants: A survey. Electr. J. Combin., Dynamic Survey DS21, 2024. URL: https://doi.org/10.37236/2713.
  37. Pavel Valtr. On the pair-crossing number. In J. E. Goodman, J. Pach, and E. Welzl, editors, Combinatorial and Computational Geometry, volume 52 of MSRI Publications, pages 569-575. Cambridge University Press, 2005. URL: https://library2.msri.org/books/Book52/files/31valtr.pdf.
  38. David R. Wood. Characterisations of intersection graphs by vertex orderings. Australasian J. Combin., 34:261-268, 2006. URL: https://ajc.maths.uq.edu.au/pdf/34/ajc_v34_p261.pdf.
  39. David R. Wood and Jan Arne Telle. Planar decompositions and the crossing number of graphs with an excluded minor. New York J. Math., 13:117-146, 2007. URL: https://nyjm.albany.edu/j/2007/13-8.html.
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