# On Point-sets that Support Planar Graphs

3 VEGAS - Effective Geometric Algorithms for Surfaces and Visibility
Inria Nancy - Grand Est, LORIA - ALGO - Department of Algorithms, Computation, Image and Geometry
Abstract : A universal point-set supports a crossing-free drawing of any planar graph. For a planar graph with $n$ vertices, if bends on edges of the drawing are permitted, universal point-sets of size $n$ are known, but only if the bend points are in arbitrary positions. If the locations of the bend points must also be specified as part of the point set, we prove that any planar graph with $n$ vertices can be drawn on a universal set $\cal S$ of $O(n^2/\log n)$ points with at most one bend per edge and with the vertices and the bend points in $\cal S$. If two bends per edge are allowed, we show that $O(n\log n)$ points are sufficient, and if three bends per edge are allowed, $O(n)$ points are sufficient. When no bends on edges are permitted, no universal point-set of size $o(n^2)$ is known for the class of planar graphs. We show that a set of $n$ points in balanced biconvex position supports the class of maximum-degree-3 series-parallel lattices.
Type de document :
Article dans une revue
Computational Geometry, Elsevier, 2013, 43 (1), pp.29--50. 〈10.1016/j.comgeo.2012.03.003〉

Littérature citée [12 références]

https://hal.inria.fr/hal-00684510
Contributeur : Sylvain Lazard <>
Soumis le : lundi 2 avril 2012 - 12:17:43
Dernière modification le : jeudi 22 septembre 2016 - 14:31:12
Document(s) archivé(s) le : mercredi 14 décembre 2016 - 19:46:14

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Vida Dujmović, Will Evans, Sylvain Lazard, William Lenhart, Giuseppe Liotta, et al.. On Point-sets that Support Planar Graphs. Computational Geometry, Elsevier, 2013, 43 (1), pp.29--50. 〈10.1016/j.comgeo.2012.03.003〉. 〈hal-00684510〉

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