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Conference Papers Year : 2011

## On Point-sets that Support Planar Graphs

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Vida Dujmović
• Function : Author
Will Evans
• Function : Author
Sylvain Lazard
William Lenhart
• Function : Author
Giuseppe Liotta
• Function : Author
Steve Wismath
• Function : Author

#### 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.

### Dates and versions

hal-00643824 , version 1 (22-11-2011)

### Identifiers

• HAL Id : hal-00643824 , version 1

### Cite

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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