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Axis-Parallel Right Angle Crossing Graphs

Authors Patrizio Angelini , Michael A. Bekos , Julia Katheder , Michael Kaufmann , Maximilian Pfister , Torsten Ueckerdt

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Author Details

Patrizio Angelini
  • John Cabot University, Rome, Italy
Michael A. Bekos
  • Department of Mathematics, University of Ioannina, Greece
Julia Katheder
  • Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany
Michael Kaufmann
  • Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany
Maximilian Pfister
  • Wilhelm-Schickard-Institut für Informatik, Universität Tübingen, Germany
Torsten Ueckerdt
  • Institute of Theoretical Informatics, Karlsruhe Institute of Technology, Germany

Funding

The work of J. Katheder and M. Kaufmann is supported by DFG grant Ka 812-18/2.

Cite AsGet BibTex

Patrizio Angelini, Michael A. Bekos, Julia Katheder, Michael Kaufmann, Maximilian Pfister, and Torsten Ueckerdt. Axis-Parallel Right Angle Crossing Graphs. In 31st Annual European Symposium on Algorithms (ESA 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 274, pp. 9:1-9:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.ESA.2023.9

Abstract

A RAC graph is one admitting a RAC drawing, that is, a polyline drawing in which each crossing occurs at a right angle. Originally motivated by psychological studies on readability of graph layouts, RAC graphs form one of the most prominent graph classes in beyond planarity. In this work, we study a subclass of RAC graphs, called axis-parallel RAC (or apRAC, for short), that restricts the crossings to pairs of axis-parallel edge-segments. apRAC drawings combine the readability of planar drawings with the clarity of (non-planar) orthogonal drawings. We consider these graphs both with and without bends. Our contribution is as follows: (i) We study inclusion relationships between apRAC and traditional RAC graphs. (ii) We establish bounds on the edge density of apRAC graphs. (iii) We show that every graph with maximum degree 8 is 2-bend apRAC and give a linear time drawing algorithm. Some of our results on apRAC graphs also improve the state of the art for general RAC graphs. We conclude our work with a list of open questions and a discussion of a natural generalization of the apRAC model.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Computational geometry
Keywords
  • Graph drawing
  • RAC graphs
  • Graph drawing algorithms

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References

  1. Pankaj K. Agarwal, Boris Aronov, János Pach, Richard Pollack, and Micha Sharir. Quasi-planar graphs have a linear number of edges. Comb., 17(1):1-9, 1997. URL: https://doi.org/10.1007/BF01196127.
  2. Patrizio Angelini, Michael A. Bekos, Henry Förster, and Michael Kaufmann. On RAC drawings of graphs with one bend per edge. Theor. Comput. Sci., 828-829:42-54, 2020. URL: https://doi.org/10.1016/j.tcs.2020.04.018.
  3. Patrizio Angelini, Michael A. Bekos, Julia Katheder, Michael Kaufmann, and Maximilian Pfister. RAC drawings of graphs with low degree. In Stefan Szeider, Robert Ganian, and Alexandra Silva, editors, MFCS, volume 241 of LIPIcs, pages 11:1-11:15. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2022. URL: https://doi.org/10.4230/LIPIcs.MFCS.2022.11.
  4. Patrizio Angelini, Michael A. Bekos, Julia Katheder, Michael Kaufmann, Maximilian Pfister, and Torsten Ueckerdt. Axis-parallel right angle crossing graphs. CoRR, abs/2306.17073, 2023. URL: https://doi.org/10.48550/arXiv.2306.17073.
  5. 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.
  6. Evmorfia N. Argyriou, Michael A. Bekos, Michael Kaufmann, and Antonios Symvonis. Geometric RAC simultaneous drawings of graphs. J. Graph Algorithms Appl., 17(1):11-34, 2013. URL: https://doi.org/10.7155/jgaa.00282.
  7. Evmorfia N. Argyriou, Michael A. Bekos, and Antonios Symvonis. The straight-line RAC drawing problem is NP-hard. J. Graph Algorithms Appl., 16(2):569-597, 2012. URL: https://doi.org/10.7155/jgaa.00274.
  8. Karin Arikushi, Radoslav Fulek, Balázs Keszegh, Filip Moric, and Csaba D. Tóth. Graphs that admit right angle crossing drawings. Comput. Geom., 45(4):169-177, 2012. URL: https://doi.org/10.1016/j.comgeo.2011.11.008.
  9. Sang Won Bae, Jean-François Baffier, Jinhee Chun, Peter Eades, Kord Eickmeyer, Luca Grilli, Seok-Hee Hong, Matias Korman, Fabrizio Montecchiani, Ignaz Rutter, and Csaba D. Tóth. Gap-planar graphs. Theor. Comput. Sci., 745:36-52, 2018. URL: https://doi.org/10.1016/j.tcs.2018.05.029.
  10. Michael A. Bekos, Walter Didimo, Giuseppe Liotta, Saeed Mehrabi, and Fabrizio Montecchiani. On RAC drawings of 1-planar graphs. Theor. Comput. Sci., 689:48-57, 2017. URL: https://doi.org/10.1016/j.tcs.2017.05.039.
  11. Michael A. Bekos, Thomas C. van Dijk, Philipp Kindermann, and Alexander Wolff. Simultaneous drawing of planar graphs with right-angle crossings and few bends. J. Graph Algorithms Appl., 20(1):133-158, 2016. URL: https://doi.org/10.7155/jgaa.00388.
  12. Therese C. Biedl and Goos Kant. A better heuristic for orthogonal graph drawings. Comput. Geom., 9(3):159-180, 1998. URL: https://doi.org/10.1016/S0925-7721(97)00026-6.
  13. Franz J. Brandenburg, Walter Didimo, William S. Evans, Philipp Kindermann, Giuseppe Liotta, and Fabrizio Montecchiani. Recognizing and drawing IC-planar graphs. Theor. Comput. Sci., 636:1-16, 2016. URL: https://doi.org/10.1016/j.tcs.2016.04.026.
  14. Steven Chaplick, Henry Förster, Myroslav Kryven, and Alexander Wolff. Drawing graphs with circular arcs and right-angle crossings. In Susanne Albers, editor, 17th Scandinavian Symposium and Workshops on Algorithm Theory, SWAT 2020, June 22-24, 2020, Tórshavn, Faroe Islands, volume 162 of LIPIcs, pages 21:1-21:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.SWAT.2020.21.
  15. Steven Chaplick, Fabian Lipp, Alexander Wolff, and Johannes Zink. Compact drawings of 1-planar graphs with right-angle crossings and few bends. Comput. Geom., 84:50-68, 2019. URL: https://doi.org/10.1016/j.comgeo.2019.07.006.
  16. Hooman Reisi Dehkordi and Peter Eades. Every outer-1-plane graph has a right angle crossing drawing. Int. J. Comput. Geom. Appl., 22(6):543-558, 2012. URL: https://doi.org/10.1142/S021819591250015X.
  17. Emilio Di Giacomo, Walter Didimo, Peter Eades, and Giuseppe Liotta. 2-layer right angle crossing drawings. Algorithmica, 68(4):954-997, 2014. URL: https://doi.org/10.1007/s00453-012-9706-7.
  18. Emilio Di Giacomo, Walter Didimo, Luca Grilli, Giuseppe Liotta, and Salvatore Agostino Romeo. Heuristics for the maximum 2-layer RAC subgraph problem. Comput. J., 58(5):1085-1098, 2015. URL: https://doi.org/10.1093/comjnl/bxu017.
  19. Emilio Di Giacomo, Walter Didimo, Giuseppe Liotta, and Henk Meijer. Area, curve complexity, and crossing resolution of non-planar graph drawings. Theory Comput. Syst., 49(3):565-575, 2011. URL: https://doi.org/10.1007/s00224-010-9275-6.
  20. Walter Didimo. Right angle crossing drawings of graphs. In Seok-Hee Hong and Takeshi Tokuyama, editors, Beyond Planar Graphs, pages 149-169. Springer, 2020. URL: https://doi.org/10.1007/978-981-15-6533-5_9.
  21. Walter Didimo, Peter Eades, and Giuseppe Liotta. A characterization of complete bipartite RAC graphs. Inf. Process. Lett., 110(16):687-691, 2010. URL: https://doi.org/10.1016/j.ipl.2010.05.023.
  22. Walter Didimo, Peter Eades, and Giuseppe Liotta. Drawing graphs with right angle crossings. Theor. Comput. Sci., 412(39):5156-5166, 2011. URL: https://doi.org/10.1016/j.tcs.2011.05.025.
  23. Peter Eades and Giuseppe Liotta. Right angle crossing graphs and 1-planarity. Discret. Appl. Math., 161(7-8):961-969, 2013. URL: https://doi.org/10.1016/j.dam.2012.11.019.
  24. Peter Eades, Antonios Symvonis, and Sue Whitesides. Three-dimensional orthogonal graph drawing algorithms. Discret. Appl. Math., 103(1-3):55-87, 2000. URL: https://doi.org/10.1016/S0166-218X(00)00172-4.
  25. Henry Förster and Michael Kaufmann. On compact RAC drawings. In Fabrizio Grandoni, Grzegorz Herman, and Peter Sanders, editors, ESA, volume 173 of LIPIcs, pages 53:1-53:21. Schloss Dagstuhl - Leibniz-Zentrum für Informatik, 2020. URL: https://doi.org/10.4230/LIPIcs.ESA.2020.53.
  26. Ulrich Fößmeier and Michael Kaufmann. Drawing high degree graphs with low bend numbers. In Franz-Josef Brandenburg, editor, GD, volume 1027 of LNCS, pages 254-266. Springer, 1995. URL: https://doi.org/10.1007/BFb0021809.
  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. F. Hadlock. Finding a maximum cut of a planar graph in polynomial time. SIAM J. Comput., 4(3):221-225, 1975. Google Scholar
  29. Seok-Hee Hong and Takeshi Tokuyama, editors. Beyond Planar Graphs. Springer, 2020. URL: https://doi.org/10.1007/978-981-15-6533-5.
  30. Weidong Huang, Peter Eades, and Seok-Hee Hong. Larger crossing angles make graphs easier to read. J. Vis. Lang. Comput., 25(4):452-465, 2014. URL: https://doi.org/10.1016/j.jvlc.2014.03.001.
  31. János Pach and Géza Tóth. Graphs drawn with few crossings per edge. Comb., 17(3):427-439, 1997. URL: https://doi.org/10.1007/BF01215922.
  32. Roberto Tamassia, editor. Handbook on Graph Drawing and Visualization. Chapman and Hall/CRC, 2013. URL: https://www.crcpress.com/Handbook-of-Graph-Drawing-and-Visualization/Tamassia/9781584884125.
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