On Point-sets that Support Planar Graphs

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, $\Theta(n)$ 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.
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Submitted on : Tuesday, November 22, 2011 - 7:39:00 PM
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Vida Dujmović, Will Evans, Sylvain Lazard, William Lenhart, Giuseppe Liotta, et al.. On Point-sets that Support Planar Graphs. 19th International Symposium on Graph Drawing, Sep 2011, Eindhoven, Netherlands. ⟨hal-00643824⟩

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