Abstract
We study intersection graphs of segments with prescribed slopes in the plane. A sufficient and necessary condition on tuples of slopes in order to define the same class of graphs is presented for both the possibilities that the parallel segments can or cannot overlap. Classes of intersection graphs of segments with four slopes are fully described; in particular, we find an infinite set of quadruples of slopes which define mutually distinct classes of intersection graphs of segments with those slopes.
Supported in part by KONTAKT ME337/99
This research was supported by GAUK 158/99 and GAČR 201/99/0242. Institute for Theoretical Computer Science (ITI) is supported as project LN00A056 by the Ministry of Education of Czech Republic.
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
T. Asano: Difficulty of the maximum independent set problem on intersection graphs of geometric objects, Graph theory, combinatorics and applications, vol.1, Wiley-Intersci.Publ., 1991, pp.9–18.
A. Bouchet: Reducing prime graphs and recognizing circle graphs, Combinatorica 7, 1987, pp.243–254.
N.de Castro, F.J. Cobos, J.C. Dana, A. Marquez, M. Noy: Triangle-free planar graphs as segments intersection graphs, J. Krato chvil (ed.), Graph drawing, 7th international symposium, Štiřýn Castle, Czech Republic, proceedings, Springer LNCS 1731, 1999, pp.341–350.
G. Ehrlich, S. Even, R.E. Tarjan: Intersection graphs of curves in the plane, J. Combinatorial Theory Ser.B 21, 1976, no.1, 8–20.
J. C. Fournier: Une caracterization des graphes de cordes, C.R. Acad. Sci. Paris 286A, 1978, pp.811–813.
H. de Fraysseix: A characterization of circle graphs, European Journal of Combinatorics 5, 1984, pp.223–238.
H.de Fraysseix, P. Ossona de Mendez, J. Pach: Representation of planar graphs by segments, Intuitive Geometry 63, 1991, pp.109–117.
M. Goljan, J. Kratochvíl, P. Kučera: String graphs, Academia, Prague 1986.
I. B.-A. Hartman, I. Newman, R. Ziv: On grid intersection graphs, Discrete Math. 87, 1991, no.1, pp.41–52.
V. B. Kalinin: On intersection graphs, Algorithmic constructions and their efficiency (in Russian), Yaroslav. Gos. Univ., 1983, pp.72–76.
S. Klavžar, M. Petkovšek: Intersection graphs of halflines and halfplanes, Discrete Math. 66, 1987, no.1–2, pp.133–137.
J. Kratochvíl: personal comunication.
J. Kratochvíl, J. Matoušek: Intersection Graphs of Segments, Journal of Combinatorial Theory, Series B, Vol.62, No.2, 1994, pp.289–315.
J. Kratochvíl, J. Matoušek: NP-hardness results for intersection graphs, Comment. Math.Univ.Carolin. 30, 1989, pp.761–773.
J. Kratochvíl, J. Nešetřil: Independent set and clique problems in intersection defined classes of graphs, Comment.Math.Univ. Carolin. 31, 1990, pp.85–93.
A. C. Tucker: An algorithm for circular-arc graphs, SIAM J.Computing 31.2, 1980, pp.211–216.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2002 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Černý, J., Král, D., Nyklová, H., Pangrác, O. (2002). On Intersection Graphs of Segments with Prescribed Slopes. In: Mutzel, P., Jünger, M., Leipert, S. (eds) Graph Drawing. GD 2001. Lecture Notes in Computer Science, vol 2265. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-45848-4_21
Download citation
DOI: https://doi.org/10.1007/3-540-45848-4_21
Published:
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-43309-5
Online ISBN: 978-3-540-45848-7
eBook Packages: Springer Book Archive