Abstract
This paper starts the investigation of a constrained version of the point-set embeddability problem. Let G = (V,E) be a planar graph with n vertices, G′ = (V′,E′) a subgraph of G, and S a set of n distinct points in the plane. We study the problem of computing a point-set embedding of G on S subject to the constraint that G′ is drawn with straight-line edges. Different drawing algorithms are presented that guarantee small curve complexity of the resulting drawing, i.e. a small number of bends per edge. It is proved that: (i) If G′ is an outerplanar graph and S is any set of points in convex position, a point-set embedding of G on S can be computed such that the edges of E ∖ E′ have at most 4 bends each. (ii) If S is any set of points in general position and G′ is a face of G or if it is a simple path, the curve complexity of the edges of E ∖ E′ is at most 8. (iii) If S is in general position and G′ is a set of k disjoint paths, the curve complexity of the edges of E ∖ E′ is O(2k).
Research partially supported by the MIUR Project “MAINSTREAM: Algorithms for massive information structures and data streams” and by NSERC.
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., Di Giacomo, E., Liotta, G.: Drawing colored graphs on colored points. Theoret. Comput. Sci. 408(2-3), 129–142 (2008)
Bose, P.: On embedding an outer-planar graph on a point set. Comput. Geom. Theory Appl. 23, 303–312 (2002)
Bose, P., McAllister, M., Snoeyink, J.: Optimal algorithms to embed trees in a point set. J. Graph Algorithms Appl. 2(1), 1–15 (1997)
Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Trotta, F., Wismath, S.K.: k-colored point-set embeddability of outerplanar graphs. J. Graph Algorithms Appl. 11(1), 29–49 (2008)
Di Giacomo, E., Liotta, G., Trotta, F.: On embedding a graph on two sets of points. Internat. J. Found. Comput. Sci. 17(5), 1071–1094 (2006)
Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Wismath, S.K.: Point-set embeddings of trees with edge constraints. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 113–124. Springer, Heidelberg (2008)
Di Giacomo, E., Didimo, W., Liotta, G., Wismath, S.K.: Curve-constrained drawings of planar graphs. Comput. Geom. Theory Appl. 30(1), 1–23 (2005)
Di Giacomo, E., Liotta, G., Trotta, F.: Drawing colored graphs with constrained vertex positions and few bends per edge. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, pp. 315–326. Springer, Heidelberg (2008)
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)
Halton, J.H.: On the thickness of graphs of given degree. Inform. Sci. 54, 219–238 (1991)
Ikebe, Y., Perles, M., Tamura, A., Tokunaga, S.: The rooted tree embedding problem into points in the plane. Discrete Comput. Geom. 11, 51–63 (1994)
Kaneko, A., Kano, M.: Straight line embeddings of rooted star forests in the plane. Discrete Appl. Math. 101, 167–175 (2000)
Kaneko, A., Kano, M.: Semi-balanced partitions of two sets of points and embeddings of rooted forests. Internat. J. Comput. Geom. Appl. 15(3), 229–238 (2005)
Kaufmann, M., Wiese, R.: Embedding vertices at points: Few bends suffice for planar graphs. J. Graph Algorithms Appl. 6(1), 115–129 (2002)
Pach, J., Törőcsik, J.: Layout of rooted trees. DIMACS Series in Discrete Math. and Theoretical Comput. Sci. 9, 131–137 (1993)
Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17, 717–728 (2001)
Sugiyama, K.: Graph Drawing and Applications for Software and Knowledge Engineers. World Scientific, Singapore (2002)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2009 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Di Giacomo, E., Didimo, W., Liotta, G., Meijer, H., Wismath, S. (2009). Constrained Point-Set Embeddability of Planar Graphs. In: Tollis, I.G., Patrignani, M. (eds) Graph Drawing. GD 2008. Lecture Notes in Computer Science, vol 5417. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-00219-9_35
Download citation
DOI: https://doi.org/10.1007/978-3-642-00219-9_35
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-642-00218-2
Online ISBN: 978-3-642-00219-9
eBook Packages: Computer ScienceComputer Science (R0)