Untangling a Planar Graph

Details Untangling a Planar Graph Untangling a Planar Graph

2007-09-03

arxiv.org/abs/0709.0170v5

Computer Science

Computational Geometry

(v5) Minor, mostly linguistic changes

Scientific paper

10.1007/s00454-008-9130-6

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.

Affiliated with

Also associated with

No associations

Rating

Untangling a Planar Graph does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Untangling a Planar Graph, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Untangling a Planar Graph will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-467930