Untangling polygons and graphs

TL;DR

This paper introduces algorithms and bounds for untangling planar graphs, specifically cycles and 3-connected planar graphs, by minimizing vertex movement while maintaining planarity.

Contribution

It presents an algorithm for untangling cycle graphs with many fixed vertices and establishes upper bounds for fixed vertices in general graphs based on structural properties.

Findings

Cycle graph C_n can be untangled with at least (n^{2/3}) vertices fixed.

Upper bounds on fixed vertices depend on graph degree and diameter.

For 3-connected planar graphs, fixed vertices are bounded by O((n log n)^{2/3}).

Abstract

Untangling is a process in which some vertices 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 at least \Omega(n^{2/3}) vertices fixed. For any 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 of its consequences is the upper bound O((n log n)^{2/3}) for all 3-vertex-connected planar graphs.