Untangling a Planar Graph
Xavier Goaoc, Jan Kratochvil, Yoshio Okamoto, Chan-Su Shin, and Andreas Spillner, Alexander Wolff

TL;DR
This paper studies the computational complexity of untangling planar graph drawings by vertex moves, proving NP-hardness, and provides algorithms with bounds on the number of vertices that can be fixed during untangling.
Contribution
It establishes NP-hardness for computing and approximating the minimum vertex moves needed to untangle planar graphs and introduces algorithms with near-optimal fixing bounds for outerplanar graphs.
Findings
Computing shift(G,δ) is NP-hard and hard to approximate.
Provided an algorithm fixing at least √((log n)-1)/log log n) vertices.
For outerplanar graphs, the algorithm fixes at least √(n/2) vertices.
Abstract
A straight-line drawing of a planar graph need not be plane, but can be made so by \emph{untangling} it, that is, by moving some of the vertices of . Let shift denote the minimum number of vertices that need to be moved to untangle . We show that shift 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 to be the maximum number of vertices of a planar -vertex graph that can be fixed when untangling . We give an algorithm that fixes at least vertices when untangling a drawing of an -vertex graph . If is outerplanar, the same algorithm fixes at least vertices. On the other hand we construct, for arbitrarily large…
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