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On RAC Drawings of Graphs with One Bend per Edge

  • Patrizio AngeliniEmail author
  • Michael A. Bekos
  • Henry Förster
  • Michael Kaufmann
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11282)

Abstract

A k-bend right-angle-crossing drawing (or k-bend RAC drawing, for short) of a graph is a polyline drawing where each edge has at most k bends and the angles formed at the crossing points of the edges are \(90^\circ \). Accordingly, a graph that admits a \(k\)-bend RAC drawing is referred to as k-bend right-angle-crossing graph (or k-bend RAC, for short). In this paper, we continue the study of the maximum edge-density of \(1\)-bend RAC graphs. We show that an n-vertex \(1\)-bend RAC graph cannot have more than \(5.5n-O(1)\) edges. We also demonstrate that there exist infinitely many n-vertex \(1\)-bend RAC graphs with exactly \(5n-O(1)\) edges. Our results improve both the previously known best upper bound of \(6.5n-O(1)\) edges and the corresponding lower bound of \(4.5n-O(\sqrt{n})\) edges by Arikushi et al. (Comput. Geom. 45(4), 169–177 (2012)).

Notes

Acknowlegdment

This project was supported by DFG grant KA812/18-1.

References

  1. 1.
    Ackerman, E.: On the maximum number of edges in topological graphs with no four pairwise crossing edges. Discrete Comput. Geom. 41(3), 365–375 (2009)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ackerman, E.: On topological graphs with at most four crossings per edge. CoRR abs/1509.01932 (2015)Google Scholar
  3. 3.
    Ackerman, E., Keszegh, B., Vizer, M.: On the size of planarly connected crossing graphs. J. Graph Algorithms Appl. 22(1), 11–22 (2018)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Ackerman, E., Tardos, G.: On the maximum number of edges in quasi-planar graphs. J. Comb. Theory, Series A 114(3), 563–571 (2007)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Agarwal, P.K., Aronov, B., Pach, J., Pollack, R., Sharir, M.: Quasi-planar graphs have a linear number of edges. Combinatorica 17(1), 1–9 (1997)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Angelini, P., Bekos, M., Förster, H., Kaufmann, M.: On RACdrawings of graphs with one bend per edge. CoRR 1808.10470 (2018). http://arxiv.org/abs/1808.10470
  7. 7.
    Angelini, P., et al.: On the perspectives opened by right angle crossing drawings. J. Graph Algorithms Appl. 15(1), 53–78 (2011)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Argyriou, E.N., Bekos, M.A., Symvonis, A.: The straight-line RAC drawing problem is NP-hard. J. Graph Algorithms Appl. 16(2), 569–597 (2012)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Arikushi, K., Fulek, R., Keszegh, B., Moric, F., Tóth, C.D.: Graphs that admit right angle crossing drawings. Comput. Geom. 45(4), 169–177 (2012)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Bachmaier, C., Brandenburg, F.J., Hanauer, K., Neuwirth, D., Reislhuber, J.: NIC-planar graphs. Discrete Appl. Math. 232, 23–40 (2017)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Bae, S.W., et al.: Gap-planar graphs. In: Frati, F., Ma, K.-L. (eds.) GD 2017. LNCS, vol. 10692, pp. 531–545. Springer, Cham (2018).  https://doi.org/10.1007/978-3-319-73915-1_41CrossRefGoogle Scholar
  12. 12.
    Bekos, M.A., Didimo, W., Liotta, G., Mehrabi, S., Montecchiani, F.: On RAC drawings of 1-planar graphs. Theor. Comput. Sci. 689, 48–57 (2017)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Bekos, M.A., Kaufmann, M., Raftopoulou, C.N.: On the density of non-simple 3-planar graphs. In: Hu, Y., Nöllenburg, M. (eds.) GD 2016. LNCS, vol. 9801, pp. 344–356. Springer, Cham (2016).  https://doi.org/10.1007/978-3-319-50106-2_27CrossRefGoogle Scholar
  14. 14.
    Bekos, M.A., Kaufmann, M., Raftopoulou, C.N.: On optimal 2- and 3-planar graphs. In: Aronov, B., Katz, M.J. (eds.) Symposium on Computational Geometry. LIPIcs, vol. 77, pp. 16:1–16:16, Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik (2017)Google Scholar
  15. 15.
    Brandenburg, F.J., Didimo, W., Evans, W.S., Kindermann, P., Liotta, G., Montecchiani, F.: Recognizing and drawing IC-planar graphs. Theor. Comput. Sci. 636, 1–16 (2016)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Chaplick, S., Lipp, F., Wolff, A., Zink, J.: 1-bend RAC drawings of NIC-planar graphs in quadratic area. In: Mulzer, W. (ed.) Proceedings of the 34th European Workshop on Computational Geometry (EuroCG 2018). Berlin (2018, to appear)Google Scholar
  17. 17.
    Cheong, O., Har-Peled, S., Kim, H., Kim, H.: On the number of edges of fan-crossing free graphs. Algorithmica 73(4), 673–695 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Di Giacomo, E., Didimo, W., Eades, P., Liotta, G.: 2-layer right angle crossing drawings. Algorithmica 68(4), 954–997 (2014)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H.: Area, curve complexity, and crossing resolution of non-planar graph drawings. Theory Comput. Syst. 49(3), 565–575 (2011)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Didimo, W., Eades, P., Liotta, G.: A characterization of complete bipartite RAC graphs. Inf. Process. Lett. 110(16), 687–691 (2010)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Didimo, W., Eades, P., Liotta, G.: Drawing graphs with right angle crossings. Theor. Comput. Sci. 412(39), 5156–5166 (2011)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Didimo, W., Liotta, G., Montecchiani, F.: A survey on graph drawing beyond planarity. CoRR abs/1804.07257 (2018)Google Scholar
  23. 23.
    Eades, P., Liotta, G.: Right angle crossing graphs and 1-planarity. Discrete Appl. Math. 161(7–8), 961–969 (2013)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Fox, J., Pach, J., Suk, A.: The number of edges in k-quasi-planar graphs. SIAM J. Discrete Math. 27(1), 550–561 (2013)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Hong, S.-H., Nagamochi, H.: Testing full outer-2-planarity in linear time. In: Mayr, E.W. (ed.) WG 2015. LNCS, vol. 9224, pp. 406–421. Springer, Heidelberg (2016).  https://doi.org/10.1007/978-3-662-53174-7_29CrossRefGoogle Scholar
  26. 26.
    Hong, S., Tokuyama, T.: Algorithmics for beyond planar graphs. NII Shonan Meeting Seminar 089, 27 November–1 December 2016Google Scholar
  27. 27.
    Huang, W.: Using eye tracking to investigate graph layout effects. In: Hong, S., Ma, K. (eds.) APVIS, pp. 97–100. IEEE Computer Society (2007)Google Scholar
  28. 28.
    Huang, W., Eades, P., Hong, S.: Larger crossing angles make graphs easier to read. J. Vis. Lang. Comput. 25(4), 452–465 (2014)CrossRefGoogle Scholar
  29. 29.
    Kaufmann, M., Kobourov, S., Pach, J., Hong, S.: Beyond planargraphs: Algorithmics and combinatorics. Dagstuhl Seminar 16452, 6-11 November 2016Google Scholar
  30. 30.
    Kaufmann, M., Ueckerdt, T.: The density of fan-planar graphs. CoRR abs/1403.6184 (2014)Google Scholar
  31. 31.
    Liotta, G.: Graph drawing beyond planarity: Some results and open problems. SoCG Week, Invited talk (4 July 2017)Google Scholar
  32. 32.
    Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Improving the crossing lemma by finding more crossings in sparse graphs. Discrete Comput. Geom. 36(4), 527–552 (2006)MathSciNetCrossRefGoogle Scholar
  33. 33.
    Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)MathSciNetCrossRefGoogle Scholar
  34. 34.
    Purchase, H.C.: Effective information visualisation: a study of graph drawing aesthetics and algorithms. Interact. Comput. 13(2), 147–162 (2000)CrossRefGoogle Scholar
  35. 35.
    Purchase, H.C., Carrington, D.A., Allder, J.: Empirical evaluation of aesthetics-based graph layout. Empir. Softw. Eng. 7(3), 233–255 (2002)CrossRefGoogle Scholar
  36. 36.
    Ringel, G.: Ein Sechsfarbenproblem auf der Kugel. Abh. Math. Sem. Univ. Hamb. 29, 107–117 (1965)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2018

Authors and Affiliations

  • Patrizio Angelini
    • 1
    Email author
  • Michael A. Bekos
    • 1
  • Henry Förster
    • 1
  • Michael Kaufmann
    • 1
  1. 1.Wilhelm-Schickhard-Institut für Informatik, Universität TübingenTübingenGermany

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