# Graphs with Large Total Angular Resolution

• Oswin Aichholzer
• Matias Korman
• Yoshio Okamoto
• Daniel Perz
• André van Renssen
• Birgit Vogtenhuber
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11904)

## Abstract

The total angular resolution of a straight-line drawing is the minimum angle between two edges of the drawing. It combines two properties contributing to the readability of a drawing: the angular resolution, which is the minimum angle between incident edges, and the crossing resolution, which is the minimum angle between crossing edges. We consider the total angular resolution of a graph, which is the maximum total angular resolution of a straight-line drawing of this graph. We prove that, up to a finite number of well specified exceptions of constant size, the number of edges of a graph with n vertices and a total angular resolution greater than $$60^{\circ }$$ is bounded by $$2n-6$$. This bound is tight. In addition, we show that deciding whether a graph has total angular resolution at least $$60^{\circ }$$ is NP-hard.

## Keywords

Graph drawing Total angular resolution Angular resolution Crossing resolution NP-hardness

## References

1. 1.
Ackerman, E., Tardos, G.: On the maximum number of edges in quasi-planar graphs. J. Comb. Theory Ser. A 114, 563–571 (2007).
2. 2.
Aichholzer, O., Korman, M., Okamoto, Y., Parada, I., Perz, D., van Renssen, A., Vogtenhuber, B.: Graphs with large total angular resolution (2019). https://arxiv.org/abs/1908.06504v1
3. 3.
Argyriou, E.N., Bekos, M.A., Symvonis, A.: The straight-line RAC drawing problem Is NP-hard. In: Černá, I., et al. (eds.) SOFSEM 2011. LNCS, vol. 6543, pp. 74–85. Springer, Heidelberg (2011).
4. 4.
Argyriou, E.N., Bekos, M.A., Symvonis, A.: Maximizing the total resolution of graphs. Comput. J. 56(7), 887–900 (2013).
5. 5.
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., Kerren, A. (eds.) GD 2018. LNCS, vol. 11282, pp. 271–285. Springer, Cham (2018).
6. 6.
Didimo, W., Eades, P., Liotta, G.: Drawing graphs with right angle crossings. Theoret. Comput. Sci. 412(39), 5156–5166 (2011).
7. 7.
Dujmovic, V., Gudmundsson, J., Morin, P., Wolle, T.: Notes on large angle crossing graphs. Chic. J. Theor. Comput. Sci. 4, 1–14 (2011).
8. 8.
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, 1035–1052 (1993).
9. 9.
Huang, W.: Using eye tracking to investigate graph layout effects. In: 2007 6th International Asia-Pacific Symposium on Visualization, pp. 97–100. IEEE (2007).
10. 10.
Huang, W., Eades, P., Hong, S.H., Lin, C.: Improving multiple aesthetics produces better graph drawings. J. Vis. Lang. Comput. 24(4), 262–272 (2013).
11. 11.
Huang, W., Hong, S.H., Eades, P.: Effects of crossing angles. In: 2008 IEEE Pacific Visualization Symposium, pp. 41–46 (2008).
12. 12.
Kreveld, M.: The quality ratio of RAC drawings and planar drawings of planar graphs. In: Brandes, U., Cornelsen, S. (eds.) GD 2010. LNCS, vol. 6502, pp. 371–376. Springer, Heidelberg (2011).