Abstract
A drawing of a graph is fan-planar if the edges intersecting a common edge a share a vertex A on the same side of a. More precisely, orienting e arbitrarily and the other edges towards A results in a consistent orientation of the crossings. So far, fan-planar drawings have only been considered in the context of simple drawings, where any two edges share at most one point, including endpoints. We show that every non-simple fan-planar drawing can be redrawn as a simple fan-planar drawing of the same graph while not introducing additional crossings. Combined with previous results on fan-planar drawings, this yields that n-vertex-graphs having such a drawing can have at most 6.5n edges and that the recognition of such graphs is NP-hard. We thereby answer an open problem posed by Kaufmann and Ueckerdt in 2014.
This work was initiated at the \(5^{th}\) DACH Workshop on Arrangements and Drawings, which was conducted online, via gathertown, in March 2021. The authors thank the organizers of the workshop for inviting us and providing a productive working atmosphere. B. K. was partially supported by DFG project WO 758/11-1. M. M. R. is supported by the Swiss National Science Foundation within the collaborative DACH project Arrangements and Drawings as SNSF Project 200021E-171681. K. K. is supported by DFG Project MU 3501/3-1 and within the Research Training Group GRK 2434 Facets of Complexity. F. S. is supported by the German Research Foundation DFG Project FE 340/12-1.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Similar content being viewed by others
Notes
- 1.
In the literature, usually more obstructions are mentioned, which we exclude for all drawings (simple or not), see Sect. 2.
- 2.
In [17], these graphs are called fan-planar. We do not use this terminology to avoid mix-ups with the class of graphs admitting (not necessarily simple) fan-planar drawings.
- 3.
More specifically, the statement and proof of [17, Lemma 1] are incorrect. A counterexample can be obtained by removing the edge g from the construction illustrated in Fig. 8 (vertices R, B correspond to the vertices u, w in [17, Lemma 1]); for a formal description of the construction see Lemma 3.
After our submission to GD’21, the authors of [17] have uploaded a new version [18] of their preprint in which they state a different definition of fan-planarity with an additional forbidden crossing configuration; also see [18, last paragraph of Sect. 1].
References
Ackerman, E., Tardos, G.: On the maximum number of edges in quasi-planar graphs. J. Comb. Theory, Ser. A 114(3), 563–571 (2007). https://doi.org/10.1016/j.jcta.2006.08.002
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). https://doi.org/10.1007/BF01196127
Angelini, P., Bekos, M.A., Kaufmann, M., Kindermann, P., Schneck, T.: 1-fan-bundle-planar drawings of graphs. Theor. Comput. Sci. 723, 23–50 (2018). https://doi.org/10.1016/j.tcs.2018.03.005
Bae, S.W., Baffier, J., Chun, J., Eades, P., Eickmeyer, K., Grilli, L., Hong, S., Korman, M., Montecchiani, F., Rutter, I., Tóth, C.D.: Gap-planar graphs. Theor. Comput. Sci. 745, 36–52 (2018). https://doi.org/10.1016/j.tcs.2018.05.029
Bekos, M.A., Cornelsen, S., Grilli, L., Hong, S.-H., Kaufmann, M.: On the recognition of fan-planar and maximal outer-fan-planar graphs. Algorithmica 79(2), 401–427 (2016). https://doi.org/10.1007/s00453-016-0200-5
Bekos, M.A., Grilli, L.: Fan-planar graphs. In: Hong, S.-H., Tokuyama, T. (eds.) Beyond Planar Graphs, pp. 131–148. Springer, Singapore (2020). https://doi.org/10.1007/978-981-15-6533-5_8
Bekos, M.A., Kaufmann, M., Raftopoulou, C.N.: On optimal 2- and 3-planar graphs. In: Aronov, B., Katz, M.J. (eds.) 33rd International Symposium on Computational Geometry, SoCG 2017, July 4–7, 2017, Brisbane, Australia. LIPIcs, vol. 77, pp. 16:1–16:16. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2017). https://doi.org/10.4230/LIPIcs.SoCG.2017.16
Binucci, C., Chimani, M., Didimo, W., Gronemann, M., Klein, K., Kratochvíl, J., Montecchiani, F., Tollis, I.G.: Algorithms and characterizations for 2-layer fan-planarity: from caterpillar to stegosaurus. J. Graph Algorithms Appl. 21(1), 81–102 (2017). https://doi.org/10.7155/jgaa.00398
Binucci, C., Giacomo, E.D., Didimo, W., Montecchiani, F., Patrignani, M., Symvonis, A., Tollis, I.G.: Fan-planarity: properties and complexity. Theor. Comput. Sci. 589, 76–86 (2015). https://doi.org/10.1016/j.tcs.2015.04.020
Brandenburg, F.J.: On fan-crossing graphs. Theor. Comput. Sci. 841, 39–49 (2020). https://doi.org/10.1016/j.tcs.2020.07.002
Cheong, O., Har-Peled, S., Kim, H., Kim, H.-S.: On the number of edges of fan-crossing free graphs. Algorithmica 73(4), 673–695 (2014). https://doi.org/10.1007/s00453-014-9935-z
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
Didimo, W., Liotta, G., Montecchiani, F.: A survey on graph drawing beyond planarity. ACM Comput. Surv. 52(1), 4:1-4:37 (2019). https://doi.org/10.1145/3301281
Holten, D.: Hierarchical edge bundles: visualization of adjacency relations in hierarchical data. IEEE Trans. Vis. Comput. Graph. 12(5), 741–748 (2006). https://doi.org/10.1109/TVCG.2006.147
Holten, D., van Wijk, J.J.: Force-directed edge bundling for graph visualization. Comput. Graph. Forum 28(3), 983–990 (2009). https://doi.org/10.1111/j.1467-8659.2009.01450.x
Hong, S.-H., Tokuyama, T. (eds.): Beyond Planar Graphs. Springer, Singapore (2020). https://doi.org/10.1007/978-981-15-6533-5
Kaufmann, M., Ueckerdt, T.: The density of fan-planar graphs. CoRR abs/1403.6184v1 (2014). http://arxiv.org/abs/1403.6184v1
Kaufmann, M., Ueckerdt, T.: The density of fan-planar graphs. CoRR abs/1403.6184v2 (2014). http://arxiv.org/abs/1403.6184v2
Klemz, B., Knorr, K., Reddy, M.M., Schröder, F.: Simplifying non-simple fan-planar drawings. CoRR abs/2108.13345 (2021). https://arxiv.org/abs/2108.13345
Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997). https://doi.org/10.1007/BF01215922
Telea, A.C., Ersoy, O.: Image-based edge bundles: simplified visualization of large graphs. Comput. Graph. Forum 29(3), 843–852 (2010). https://doi.org/10.1111/j.1467-8659.2009.01680.x
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 Springer Nature Switzerland AG
About this paper
Cite this paper
Klemz, B., Knorr, K., Reddy, M.M., Schröder, F. (2021). Simplifying Non-simple Fan-Planar Drawings. In: Purchase, H.C., Rutter, I. (eds) Graph Drawing and Network Visualization. GD 2021. Lecture Notes in Computer Science(), vol 12868. Springer, Cham. https://doi.org/10.1007/978-3-030-92931-2_4
Download citation
DOI: https://doi.org/10.1007/978-3-030-92931-2_4
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-92930-5
Online ISBN: 978-3-030-92931-2
eBook Packages: Computer ScienceComputer Science (R0)