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Let $G=(V,E)$ be a planar graph, only specified by its set of vertices and edges. Suppose $|V|=n$. According to Fary's theorem, there exists a planar embedding of $G$ with straight line segments. Then this embedding can be represented by a set of points in the plane (each $v\in V$ is represented by a unique point $p_v$ and there is a line segment between $p_u$ and $p_v$ iff $(u,v)\in E$)

Questions :

  1. Is it possible to find this set of points in polynomial time (i.e. $n^{O(1)}$) ?
  2. What if we force the points to have integer coordinates ?
  3. Can we force the maximum (euclidian) distance between any two points to be $n^{O(1)}$ ?

Such statements could be very useful for proving lower bounds for some geometrical problems

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  • $\begingroup$ All three of your questions are answered in the page you link. $\endgroup$ – Peter Taylor Jun 20 '18 at 14:50
  • $\begingroup$ I'm not sure, in this article they never mention the complexity of computing this planar embedding in polynomial time. My questions are not about existence but time complexity $\endgroup$ – Mathieu Mari Jun 25 '18 at 12:48
  • $\begingroup$ In 'Related Results' they do mention being able to solve in linear time for a particular case. $\endgroup$ – koverman47 Jul 25 '18 at 15:08
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Quoting directly from the Wikipedia article linked in the question:

De Fraysseix, Pach and Pollack showed how to find in linear time a straight-line drawing in a grid with dimensions linear in the size of the graph

As I commented earlier, this answers all three questions.

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