Abstract
We consider the point set embedding problem (PSE) for 1-, 2- and k-planar graphs where at most 1, 2, or k crossings resp. are allowed for each edge which greatly extends the well-researched class of planar graphs. For any set of n points and any given embedded graph that belongs to one of the above graph classes, we compute a 1-to-1 mapping of the vertices to the points such that the edges can be routed using only a limited number of bends according to the given embedding and the sequences of crossings. Surprisingly, for the class of 1-planar graphs the same results can be achieved as the best known results for planar graphs. Additionally for k-planar graphs, the bounds are also much better than expected from the first sight.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Ackerman, E., Tardos, G.: On the maximum number of edges in quasi-planar graphs. J. Comb. Theor. Ser. A 114(3), 563–571 (2007)
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)
Angelini, P., et al.: Small universal point sets for k-outerplanar graphs. Discret. Comput. Geom. 60(2), 430–470 (2018). https://doi.org/10.1007/s00454-018-0009-x
Bannister, M.J., Cheng, Z., Devanny, W.E., Eppstein, D.: Superpatterns and universal point sets. J. Graph Algorithms Appl. 18(2), 177–209 (2014). https://doi.org/10.7155/jgaa.00318
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)
Biedl, T.C., Kant, G., Kaufmann, M.: On triangulating planar graphs under the four-connectivity constraint. Algorithmica 19(4), 427–446 (1997). https://doi.org/10.1007/PL00009182
Bose, P.: On embedding an outer-planar graph in a point set. In: Proceedings of Graph Drawing, 5th International Symposium, GD 1997, 18–20 September 1997, Rome, Italy, pp. 25–36 (1997). https://doi.org/10.1007/3-540-63938-1_47
Bose, P., McAllister, M., Snoeyink, J.: Optimal algorithms to embed trees in a point set. In: Proceedings of Graph Drawing, Symposium on Graph Drawing, GD 1995, 20–22 September 1995, Passau, Germany, pp. 64–75 (1995). https://doi.org/10.1007/BFb0021791
Cabello, S.: Planar embeddability of the vertices of a graph using a fixed point set is NP-hard. J. Graph Algorithms Appl. 10(2), 353–363 (2006). http://jgaa.info/accepted/2006/Cabello2006.10.2.pdf
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)
Chiba, N., Nishizeki, T.: The hamiltonian cycle problem is linear-time solvable for 4-connected planar graphs. J. Algorithms 10(2), 187–211 (1989). https://doi.org/10.1016/0196-6774(89)90012-6
Chrobak, M., Karloff, H.J.: A lower bound on the size of universal sets for planar graphs. SIGACT News 20(4), 83–86 (1989). https://doi.org/10.1145/74074.74088
Didimo, W., Eades, P., Liotta, G.: Drawing graphs with right angle crossings. Theor. Comput. Sci. 412(39), 5156–5166 (2011)
Didimo, W., Liotta, G., Montecchiani, F.: A survey on graph drawing beyond planarity. CoRR abs/1804.07257 (2018)
Feige, U.: Approximating the bandwidth via volume respecting embeddings (extended abstract). In: Proceedings of the Thirtieth Annual ACM Symposium on the Theory of Computing, 23–26 May 1998, Dallas, Texas, USA, pp. 90–99 (1998). https://doi.org/10.1145/276698.276716
Fox, J., Pach, J., Suk, A.: The number of edges in k-quasi-planar graphs. SIAM J. Discret. Math. 27(1), 550–561 (2013)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990). https://doi.org/10.1007/BF02122694
Fulek, R., Tóth, C.D.: Universal point sets for planar three-trees. J. Discret. Algorithms 30, 101–112 (2015). https://doi.org/10.1016/j.jda.2014.12.005
Hong, S., Tokuyama, T.: Algorithmics for beyond planar graphs. NII Shonan Meeting Seminar 089, 27 November–1 December 2016
Ikebe, Y., Perles, M.A., Tamura, A., Tokunaga, S.: The rooted tree embedding problem into points in the plane. Discret. Comput. Geom. 11, 51–63 (1994). https://doi.org/10.1007/BF02573994
Kaufmann, M., Kobourov, S., Pach, J., Hong, S.: Beyond planar graphs: algorithmics and combinatorics. Dagstuhl Seminar 16452, 6–11 November 2016
Kaufmann, M., Ueckerdt, T.: The density of fan-planar graphs. CoRR abs/1403.6184 (2014)
Kaufmann, M., Wiese, R.: Embedding vertices at points: few bends suffice for planar graphs. J. Graph Algorithms Appl. 6(1), 115–129 (2002). http://www.cs.brown.edu/publications/jgaa/accepted/2002/KaufmannWiese2002.6.1.pdf
Leighton, F.T.: Introduction to Parallel Algorithms and Architectures: Array, Trees, Hypercubes. Morgan Kaufmann Publishers Inc., San Francisco (1992)
Liotta, G.: Graph drawing beyond planarity: some results and open problems. SoCG Week, Invited talk, 4 July 2017
Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Improving the crossing lemma by finding more crossings in sparse graphs. Discret. Comput. Geom. 36(4), 527–552 (2006)
Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17(3), 427–439 (1997)
Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. In: Proceedings of Graph Drawing, 6th International Symposium, GD 1998, August 1998, Montréal, Canada, pp. 263–274 (1998). https://doi.org/10.1007/3-540-37623-2_20
Pach, J., Gritzmann, P., Mohar, B., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Am. Math. Mon. 98, 165–166 (1991). Professor Pach’s number: [065]
Ringel, G.: Ein Sechsfarbenproblem auf der Kugel. Abh. Math. Sem. Univ. Hamb. 29, 107–117 (1965)
Acknowledgement
The author wishes to thanks the participants of the GNV workshop in Heiligkreuztal 2018 for inspiring discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2019 Springer Nature Switzerland AG
About this paper
Cite this paper
Kaufmann, M. (2019). On Point Set Embeddings for k-Planar Graphs with Few Bends per Edge. In: Catania, B., Královič, R., Nawrocki, J., Pighizzini, G. (eds) SOFSEM 2019: Theory and Practice of Computer Science. SOFSEM 2019. Lecture Notes in Computer Science(), vol 11376. Springer, Cham. https://doi.org/10.1007/978-3-030-10801-4_21
Download citation
DOI: https://doi.org/10.1007/978-3-030-10801-4_21
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-10800-7
Online ISBN: 978-3-030-10801-4
eBook Packages: Computer ScienceComputer Science (R0)