Abstract
Let R and B be two sets of distinct points such that the points of R are coloured red and the points of B are coloured blue. Let \(\cal G\) be a family of planar graphs such that for each graph in the family |R| vertices are red and |B| vertices are blue. The set R ∪ B is a universal pointset for \(\cal G\) if every graph \(G \in \cal G\) has a straight-line planar drawing such that the blue vertices of G are mapped to the points of B and the red vertices of G are mapped to the points of R. In this paper we describe universal pointsets for meaningful classes of 2-coloured trees and show applications of these results to the coloured simultaneous geometric embeddability problem.
Research supported in part by MIUR under project AlgoDEEP prot. 2008TFBWL4.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Badent, M., Giacomo, E.D., Liotta, G.: Drawing colored graphs on colored points. Theor. Comput. Sci. 408(2-3), 129–142 (2008)
Abellanas, M., Garcia-Lopez, J., Hernández-Peñver, G., Noy, M., Ramos, P.A.: Bipartite embeddings of trees in the plane. Discrete Applied Mathematics 93(2-3), 141–148 (1999)
Brandes, U., Erten, C., Fowler, J.J., Frati, F., Geyer, M., Gutwenger, C., Hong, S.-H., Kaufmann, M., Kobourov, S.G., Liotta, G., Mutzel, P., Symvonis, A.: Colored simultaneous geometric embeddings. In: Lin, G. (ed.) COCOON 2007. LNCS, vol. 4598, pp. 254–263. Springer, Heidelberg (2007)
de Fraysseix, H., Pach, J., Pollack, R.: How to draw a planar graph on a grid. Combinatorica 10(1), 41–51 (1990)
Estrella-Balderrama, A., Fowler, J.J., Kobourov, S.G.: Colored simultaneous geometric embeddings and universal pointsets. In: Proceedings of the 21st Canadian Conference on Computational Geometry (CCCG 2009), pp. 17–20 (2009)
Everett, H., Lazard, S., Liotta, G., Wismath, S.K.: Universal sets of points for one-bend drawings of planar graphs with vertices. Discrete & Computational Geometry 43(2), 272–288 (2010)
Giacomo, E.D., Didimo, W., Liotta, G., Meijer, H., Trotta, F., Wismath, S.K.: k-colored point-set embeddability of outerplanar graphs. J. Graph Algorithms Appl. 12(1), 29–49 (2008)
Giacomo, E.D., Didimo, W., Liotta, G., Meijer, H., Wismath, S.K.: Point-set embeddings of trees with given partial drawings. Comput. Geom. 42(6-7), 664–676 (2009)
Giacomo, E.D., Liotta, G., Trotta, F.: On embedding a graph on two sets of points. Int. J. Found. Comput. Sci. 17(5), 1071–1094 (2006)
Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. Amer. Math. Monthly 98(2), 165–166 (1991)
Ikebe, Y., Perles, M.A., Tamura, A., Tokunaga, S.: The rooted tree embedding problem into points in the plane. Discrete & Computational Geometry 11, 51–63 (1994)
Kaneko, A., Kano, M.: Straight-line embeddings of two rooted trees in the plane. Discrete & Computational Geometry 21(4), 603–613 (1999)
Kaneko, A., Kano, M.: Straight line embeddings of rooted star forests in the plane. Discrete Applied Mathematics 101(1-3), 167–175 (2000)
Kanenko, A., Kano, M.: Discrete geometry on red and blue points in the plane - a survey. In: Discrete and Computational Geometry. Algorithms and Combinatories, vol. 25, pp. 551–570. Springer, Heidelberg (2003)
Kaneko, A., Kano, M.: Semi-balanced partitions of two sets of points and embeddings of rooted forests. Int. J. Comput. Geometry Appl. 15(3), 229–238 (2005)
Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. Journal of Graph Algorithms and Applications 6(1), 115–129 (2002)
Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graph and Combinatorics 17, 717–728 (2001)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2011 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
van Garderen, M., Liotta, G., Meijer, H. (2011). Universal Pointsets for 2-Coloured Trees. In: Brandes, U., Cornelsen, S. (eds) Graph Drawing. GD 2010. Lecture Notes in Computer Science, vol 6502. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-18469-7_33
Download citation
DOI: https://doi.org/10.1007/978-3-642-18469-7_33
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-18468-0
Online ISBN: 978-3-642-18469-7
eBook Packages: Computer ScienceComputer Science (R0)