Untangling a Planar Graph

Abstract

A straight-line drawing $\delta$ 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$(G,\delta )$ 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 \textsc{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 $\sqrt{((\log n)-1)/\log \log n}$ 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 ${\delta }_G$ of $G$ with fix$(G,{\delta }_G)\le \sqrt{n-2}+1$ and an $n$-vertex outerplanar graph $H$ and a drawing ${\delta }_H$ of $H$ with fix$(H,{\delta }_H)\le 2\sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs.

(v5) Minor, mostly linguistic changes