On Graphs Supported by Line Sets

Dujmovic, Vida and Evans, William and Kobourov, Stephen G. and Liotta, Giuseppe and Weibel, Christophe and Wismath, Stephen (2011) On Graphs Supported by Line Sets. In: Graph Drawing 18th International Symposium, GD 2010, September 21-24, 2010, Konstanz, Germany, , pp. 177-182 (Official URL: http://dx.doi.org/10.1007/978-3-642-18469-7_16).

Full text not available from this repository.

Abstract

For a set S of n lines labeled from 1 to n, we say that S supports an n-vertex planar graph G if for every labeling from 1 to n of its vertices, G has a straight-line crossing-free drawing with each vertex drawn as a point on its associated line. It is known from previous work [4] that no set of n parallel lines supports all n-vertex planar graphs. We show that intersecting lines, even if they intersect at a common point, are more powerful than a set of parallel lines. In particular, we prove that every such set of lines supports outerpaths, lobsters, and squids, none of which are supported by any set of parallel lines. On the negative side, we prove that no set of n lines that intersect in a common point supports all n-vertex planar graphs. Finally, we show that there exists a set of n lines in general position that does not support all n-vertex planar graphs.

Item Type:Conference Paper
Additional Information:10.1007/978-3-642-18469-7_16
Classifications:Z Theory > Z.250 Geometry
ID Code:1204

Repository Staff Only: item control page

References

Badent, M., Giacomo, E.D., Liotta, G.: Drawing colored graphs on colored points. Theor. Comput. Sci. 408(2-3), 129–142 (2008)

Bose, P.: On embedding an outer-planar graph in a point set. Computational Geometry: Theory and Applications 23(3), 303–312 (2002)

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.: Characterization of unlabeled level planar trees. Computational Geometry: Theory and Applications 42(7), 704–721 (2009)

Fowler, J.J., Kobourov, S.G.: Characterization of unlabeled level planar graphs. In: Hong, S.-H., Nishizeki, T., Quan, W. (eds.) GD 2007. LNCS, vol. 4875, Springer, Heidelberg (2008)

Gritzmann, P., Mohar, B., Pach, J., Pollack, R.: Embedding a planar triangulation with vertices at specified points. American Math. Monthly 98, 165–166 (1991)

Pach, J., Wenger, R.: Embedding planar graphs at fixed vertex locations. Graphs and Combinatorics 17, 717–728 (2001)

Rosenstiehl, P., Tarjan, R.E.: Rectilinear planar layouts and bipolar orientations of planar graphs. Discrete Computational Geometry 1(4), 343–353 (1986)