# Untangling Polygons and Graphs

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DOI: 10.1007/s00454-009-9150-x

- Cite this article as:
- Cibulka, J. Discrete Comput Geom (2010) 43: 402. doi:10.1007/s00454-009-9150-x

- 2 Citations
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## Abstract

Untangling is a process in which some vertices in a drawing of a planar graph are moved to obtain a straight-line plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph *C*_{n} while keeping Ω(*n*^{2/3}) vertices fixed.

For any connected graph *G*, we also present an upper bound on the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree, and diameter of *G*. One consequence is that every 3-connected planar graph has a drawing *δ* such that at most *O*((*n*log *n*)^{2/3}) vertices are fixed in every untangling of *δ*.