On embedding an outer-planar graph in a point set
Computational Geometry: Theory and Applications
Drawing colored graphs on colored points
Theoretical Computer Science
Characterization of unlabeled level planar trees
Computational Geometry: Theory and Applications
Characterization of unlabeled level planar graphs
GD'07 Proceedings of the 15th international conference on Graph drawing
A center transversal theorem for hyperplanes and applications to graph drawing
Proceedings of the twenty-seventh annual symposium on Computational geometry
Universal line-sets for drawing planar 3-trees
WALCOM'12 Proceedings of the 6th international conference on Algorithms and computation
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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.