Abstract
Let G be a graph without loops or multiple edges drawn in the plane. It is shown that, for any k, if G has at least C k n edges and n vertices, then it contains three sets of k edges, such that every edge in any of the sets crosses all edges in the other two sets. Furthermore, two of the three sets can be chosen such that all k edges in the set have a common vertex.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
Agarwal, P.K., Aronov, B., Pach, J., Pollack, R., Sharir, M.: Quasi-planar graphs have a linear number of edges. Combinatorica 17, 1–9 (1997)
Pach, J.: Geometric graph theory. In: Lamb, J.D., Preece, D.A. (eds.) Surveys in Combinatorics. London Mathematical Society Lecture Notes, vol. 267, pp. 167–200. Cambridge University Press, Cambridge (1999)
Pach, J., Radoičić, R., Tóth, G.: Relaxing planarity for topological graphs. In: Akiyama, J., Kano, M. (eds.) JCDCG 2002. LNCS, vol. 2866, pp. 221–232. Springer, Heidelberg (2003)
Pach, J., Radoičić, R., Tóth, G.: A generalization of quasi-planarity. In: Pach, J. (ed.) Towards a Theory of Geometric Graphs. Contemporary Mathematics, vol. 342, pp. 177–183. AMS (2004)
Pach, J., Pinchasi, R., Sharir, M., Tóth, G.: Topological graphs with no large grids, Graphs and Combinatorics Special Issue dedicated to Victor Neumann-Lara
Pach, J., Pinchasi, R., Tardos, G., Tóth, G.: Geometric graphs with no self-intersecting path of length three. In: Goodrich, M.T., Kobourov, S.G. (eds.) GD 2002. LNCS, vol. 2528, pp. 295–311. Springer, Heidelberg (2002)
Pach, J., Radoičić, R., Tardos, G., Tóth, G.: Improving the Crossing Lemma by finding more crossings in sparse graphs. In: Proceedings of the 20th Annual Symposium on Computational Geometry (SoCG 2004), pp. 76–85 (2004)
Pach, J., Tóth, G.: Graphs drawn with few crossings per edge. Combinatorica 17, 427–439 (1997)
Schaefer, M., Stefankovič, D.: Decidability of string graphs. In: Proceedings of the 33rd Annual Symposium on the Theory of Computing (STOC 2001), pp. 241–246 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Tardos, G., Tóth, G. (2005). Crossing Stars in Topological Graphs. In: Akiyama, J., Kano, M., Tan, X. (eds) Discrete and Computational Geometry. JCDCG 2004. Lecture Notes in Computer Science, vol 3742. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11589440_19
Download citation
DOI: https://doi.org/10.1007/11589440_19
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30467-8
Online ISBN: 978-3-540-32089-0
eBook Packages: Computer ScienceComputer Science (R0)