Untangling a Planar Graph
| dc.creator | Goaoc, Xavier | |
| dc.creator | Kratochvil, Jan | |
| dc.creator | Okamoto, Yoshio | |
| dc.creator | Shin, Chan-Su | |
| dc.creator | Spillner, Andreas | |
| dc.creator | Wolff, Alexander | |
| dc.date | 2007-09-03 | |
| dc.date | 2009-01-27 | |
| dc.date.accessioned | 2026-07-07T12:34:09Z | |
| dc.date.available | 2026-07-07T12:34:09Z | |
| dc.description | A straight-line drawing $δ$ 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$(G,δ)$ denote the minimum number of vertices that need to be moved to untangle $δ$. We show that shift$(G,δ)$ 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$(G,δ)=n-shift(G,δ)$ to be the maximum number of vertices of a planar $n$-vertex graph $G$ that can be fixed when untangling $δ$. 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 $δ_G$ of $G$ with fix$(G,δ_G) \le \sqrt{n-2}+1$ and an $n$-vertex outerplanar graph $H$ and a drawing $δ_H$ of $H$ with fix$(H,δ_H) \le 2 \sqrt{n-1}+1$. Thus our algorithm is asymptotically worst-case optimal for outerplanar graphs. | |
| dc.description | (v5) Minor, mostly linguistic changes | |
| dc.identifier | https://arxiv.org/abs/0709.0170 | |
| dc.identifier | http://arxiv.org/abs/0709.0170 | |
| dc.identifier | doi:10.1007/s00454-008-9130-6 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/217324 | |
| dc.subject | Computational Geometry | |
| dc.subject | Discrete Mathematics | |
| dc.subject | G.2.2; I.3.5 | |
| dc.title | Untangling a Planar Graph | |
| dc.type | text |