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Beyond Planar Graphs pp 171–186Cite as

Angular Resolutions: Around Vertices and Crossings

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Abstract

Angular resolution is one of the well-known esthetic criteria for graph drawing, but its theoretical properties are not well understood. For a straight-line drawing of a graph, its vertex angular resolution is the minimum angle formed by two consecutive edges around a vertex, and its crossing angular resolution is the minimum angle formed by a crossing, while the crossing angular resolution is defined to be \(2\pi \) if there is no crossing. The total angular resolution of a straight-line drawing is the minimum of the vertex angular resolution and the crossing angular resolution. The vertex/crossing/total angular resolution of a graph is the supremum of the vertex/crossing/total angular resolution of any straight-line drawing of the graph. In this chapter, we review some of the results on angular resolution in the literature, and identify several open problems in the field.

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Notes

  1. 1.

    This can be seen as follows. We will color the vertices of H with colors in \(\{1,2,\dots ,\varDelta (H)+1\}\). When we color a vertex v, the number of vertices that have already been colored is at most the degree of v, which is at most \(\varDelta (H)\). Thus, there is still a color that remains unused and this color can be used for v.

References

  1. Aichholzer, O., Korman, M., Okamoto, Y., Parada, I., Perz, D., van Renssen, A., Vogtenhuber, B.: Graphs with large total angular resolution. In: Archambault, D., Tóth, C.D. (eds.) Graph Drawing and Network Visualization - 27th International Symposium, GD 2019, Prague, Czech Republic, 17–20 September 2019, Proceedings. Lecture Notes in Computer Science, vol. 11904, pp. 193–199. Springer (2019). https://doi.org/10.1007/978-3-030-35802-0_15

  2. Angelini, P., Cittadini, L., Didimo, W., Frati, F., Di Battista, G., Kaufmann, M., Symvonis, A.: On the perspectives opened by right angle crossing drawings. J. Graph Algorithms Appl. 15(1), 53–78 (2011). https://doi.org/10.7155/jgaa.00217

  3. 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). https://doi.org/10.7155/jgaa.00274

  4. Argyriou, E.N., Bekos, M.A., Symvonis, A.: Maximizing the total resolution of graphs. Comput. J. 56(7), 887–900 (2013). https://doi.org/10.1093/comjnl/bxs088

  5. Barát, J., Matoušek, J., Wood, D.R.: Bounded-degree graphs have arbitrarily large geometric thickness. Electron. J. Comb. 13(1) (2006)

    Google Scholar 

  6. Bekos, M.A., Förster, H., Geckeler, C., Holländer, L., Kaufmann, M., Spallek, A.M., Splett, J.: A heuristic approach towards drawings of graphs with high crossing resolution. In: Biedl, T.C., Kerren, A. (eds.) Graph Drawing and Network Visualization - 26th International Symposium, GD 2018, Barcelona, Spain, 26–28 September 2018, Proceedings. Lecture Notes in Computer Science, vol. 11282, pp. 271–285. Springer (2018). https://doi.org/10.1007/978-3-030-04414-5_19

  7. 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). https://doi.org/10.1007/s00224-010-9275-6

  8. Di Giacomo, E., Didimo, W., Eades, P., Hong, S., Liotta, G.: Bounds on the crossing resolution of complete geometric graphs. Discret. Appl. Math. 160(1–2), 132–139 (2012). https://doi.org/10.1016/j.dam.2011.09.016

  9. Didimo, W., Eades, P., Liotta, G.: Drawing graphs with right angle crossings. Theor. Comput. Sci. 412(39), 5156–5166 (2011). https://doi.org/10.1016/j.tcs.2011.05.025

  10. Didimo, W., Kaufmann, M., Liotta, G., Okamoto, Y., Spillner, A.: Vertex angle and crossing angle resolution of leveled tree drawings. Inf. Process. Lett. 112(16), 630–635 (2012). https://doi.org/10.1016/j.ipl.2012.05.006

  11. Dillencourt, M.B., Eppstein, D., Hirschberg, D.S.: Geometric thickness of complete graphs. J. Graph Algorithms Appl. 4(3), 5–17 (2000). https://doi.org/10.7155/jgaa.00023

  12. Dujmovic, V., Gudmundsson, J., Morin, P., Wolle, T.: Notes on large angle crossing graphs. Chic. J. Theor. Comput. Sci. 2011 (2011)

    Google Scholar 

  13. Formann, M., Hagerup, T., Haralambides, J., Kaufmann, M., Leighton, F.T., Symvonis, A., Welzl, E., Woeginger, G.J.: Drawing graphs in the plane with high resolution. SIAM J. Comput. 22(5), 1035–1052 (1993). https://doi.org/10.1137/0222063

  14. Garg, A., Tamassia, R.: Planar drawings and angular resolution: algorithms and bounds (extended abstract). In: van Leeuwen, J. (ed.) Algorithms - ESA ’94, Second Annual European Symposium, Utrecht, The Netherlands, 26–28 September 1994, Proceedings. Lecture Notes in Computer Science, vol. 855, pp. 12–23. Springer (1994). https://doi.org/10.1007/BFb0049393

  15. Havet, F., van den Heuvel, J., McDiarmid, C., Reed, B.: List colouring squares of planar graphs (2008). arXiv:0807.3233

  16. van den Heuvel, J., McGuinness, S.: Coloring the square of a planar graph. J. Graph Theory 42(2), 110–124 (2003). https://doi.org/10.1002/jgt.10077

  17. Hoffmann, M., van Kreveld, M.J., Kusters, V., Rote, G.: Quality ratios of measures for graph drawing styles. In: Proceedings of the 26th Canadian Conference on Computational Geometry, CCCG 2014, Halifax, Nova Scotia, Canada, 2014. Carleton University, Ottawa, Canada (2014). http://www.cccg.ca/proceedings/2014/papers/paper05.pdf

  18. Keszegh, B., Pach, J., Pálvölgyi, D.: Drawing planar graphs of bounded degree with few slopes. SIAM J. Discret. Math. 27(2), 1171–1183 (2013). https://doi.org/10.1137/100815001

  19. Koebe, P.: Kontaktprobleme der konformen Abbildung. Ber. Sächs. Akad. Wiss. Leipzig Math.-Phys. Kl. 88, 141–164 (1936)

    Google Scholar 

  20. van Kreveld, M.J.: The quality ratio of RAC drawings and planar drawings of planar graphs. In: Brandes, U., Cornelsen, S. (eds.) Graph Drawing - 18th International Symposium, GD 2010, Konstanz, Germany, 21–24 September 2010. Revised Selected Papers. Lecture Notes in Computer Science, vol. 6502, pp. 371–376. Springer (2010). https://doi.org/10.1007/978-3-642-18469-7_34

  21. Malitz, S.M., Papakostas, A.: On the angular resolution of planar graphs. SIAM J. Discret. Math. 7(2), 172–183 (1994). https://doi.org/10.1137/S0895480193242931

  22. Mukkamala, P., Pálvölgyi, D.: Drawing cubic graphs with the four basic slopes. In: van Kreveld, M.J., Speckmann, B. (eds.) Graph Drawing - 19th International Symposium, GD 2011, Eindhoven, The Netherlands, 21–23 September 2011, Revised Selected Papers. Lecture Notes in Computer Science, vol. 7034, pp. 254–265. Springer (2011). https://doi.org/10.1007/978-3-642-25878-7_25

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Acknowledgements

The author’s research was partially supported by JSPS/MEXT KAKENHI Grant Numbers JP24106005 and JP15K00009, JST CREST Grant Number JPMJCR1402, and Kayamori Foundation of Informational Science Advancement. The author’s thanks go to many people: Seok-Hee Hong and Takeshi Tokuyama for inviting him to NII Shonan Meeting “Algorithmics for Beyond Planar Graphs” in 2016, where a part of the material of this chapter was presented; Martin Nöllenburg for his detailed comments that improved the text considerably; Michael Bekos for pointing out that the hardness proof by Formann et al. [13] implies the hardness of computing the total angular resolution; Oswin Aichholzer, Fabian Klute, Irene Parada, Daniel Perz, and Birgit Vogtenhuber for their proof of the fact that the total angular resolutions of \(Q_3\) and the Petersen graph are at most \(\pi /3\), which is an outcome from the Japan-Austrian Bilateral Seminar: Computational Geometry Seminar with Applications to Sensor Networks in 2018.

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Authors and Affiliations

  1. The University of Electro-Communications, Chofugaoka 1–5–1, Chofu, Tokyo, Japan

    Yoshio Okamoto

  2. RIKEN Center for Advanced Intelligence Project, Nihonbashi 1-chome Mitsui Building, 15th floor, Nihonbashi 1–4–1, Chuo-ku, Tokyo, Japan

    Yoshio Okamoto

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  1. Yoshio Okamoto
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Correspondence to Yoshio Okamoto .

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Editors and Affiliations

  1. School of Computer Science, University of Sydney, Sydney, NSW, Australia

    Seok-Hee Hong

  2. School of Science and Technology, Kwansei Gakuin University, Sanda, Hyogo, Japan

    Takeshi Tokuyama

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Okamoto, Y. (2020). Angular Resolutions: Around Vertices and Crossings. In: Hong, SH., Tokuyama, T. (eds) Beyond Planar Graphs. Springer, Singapore. https://doi.org/10.1007/978-981-15-6533-5_10

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  • DOI: https://doi.org/10.1007/978-981-15-6533-5_10

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