Untangling polygons and graphs

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 Cn while keeping Ω(n2/3) 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. © 2009 Springer Science+Business Media, LLC.