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