Small sets supporting fary embeddings of planar graphs
STOC '88 Proceedings of the twentieth annual ACM symposium on Theory of computing
An optimal-time algorithm for slope selection
SIAM Journal on Computing
Discrete & Computational Geometry - Selected papers from the fifth annual ACM symposium on computational geometry, Saarbrücken, Germany, June 5-11, 1989
Lectures on Discrete Geometry
STACS '03 Proceedings of the 20th Annual Symposium on Theoretical Aspects of Computer Science
On embedding an outer-planar graph in a point set
Computational Geometry: Theory and Applications
Characterization of unlabeled level planar trees
Computational Geometry: Theory and Applications
A Polynomial Bound for Untangling Geometric Planar Graphs
Discrete & Computational Geometry
On graphs supported by line sets
GD'10 Proceedings of the 18th international conference on Graph drawing
Higher-order Erdös: Szekeres theorems
Proceedings of the twenty-eighth annual symposium on Computational geometry
Ramsey-type results for semi-algebraic relations
Proceedings of the twenty-ninth annual symposium on Computational geometry
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Motivated by an open problem from graph drawing, we study several partitioning problems for line and hyperplane arrangements. We prove a ham-sandwich cut theorem: given two sets of n lines in R2, there is a line l such that in both line sets, for both halfplanes delimited by l, there are √n lines which pairwise intersect in that halfplane, and this bound is tight; a centerpoint theorem: for any set of n lines there is a point such that for any halfplane containing that point there are √n/3 of the lines which pairwise intersect in that halfplane. We generalize those results in higher dimension and obtain a center transversal theorem, a same-type lemma, and a positive portion Erdos-Szekeres theorem for hyperplane arrangements. This is done by formulating a generalization of the center transversal theorem which applies to set functions that are much more general than measures. Back to Graph Drawing (and in the plane), we completely solve the open problem that motivated our search: there is no set of n labelled lines that are universal for all n-vertex labelled planar graphs. As a side note, we prove that every set of n (unlabelled) lines is universal for all n-vertex (unlabelled) planar graphs.