A straight-line drawing $��$ of a planar graph $G$ need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of $G$. Let shift$(G,��)$ denote the minimum number of vertices that need to be moved to untangle $��$. We show that shift$(G,��)$ is NP-hard to compute and to approximate. Our hardness results extend to a version of \textsc{1BendPointSetEmbeddability}, a well-known graph-drawing problem. Further we define fix$(G,��)=n-shift(G,��)$ to be the maximum number of vertices of a planar $n$-vertex graph $G$ that can be fixed when untangling $��$. We give an algorithm that fixes at least $\sqrt{((\log n)-1)/\log \log n}$ vertices when untangling a drawing of an $n$-vertex graph $G$. If $G$ is outerplanar, the same algorithm fixes at least $\sqrt{n/2}$ vertices. On the other hand we construct, for arbitrarily large $n$, an $n$-vertex planar graph $G$ and a drawing $��_G$ of $G$ with fix$(G,��_G) \le \sqrt{n-2}+1$ and an $n$-vertex outerplanar graph $H$ and a drawing $��_H$ of $H$ with fix$(H,��_H) \le 2 \sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

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