Untangling Planar Curves

TL;DR

This paper establishes tight bounds on the number of local transformations needed to simplify planar curves and transform curve immersions, improving previous bounds and connecting topological invariants with graph transformations.

Contribution

It proves the exact asymptotic number of homotopy moves required for simplifying planar curves, improving previous bounds, and relates these results to graph transformations and surface topology.

Findings

Simplifying planar curves requires (n^{3/2}) homotopy moves in the worst case.

Transforming immersions of multiple circles requires (n^{3/2} + nk + k^2) moves.

Transforming noncontractible curves on surfaces requires (n^2) moves, tight for curves homotopic to simple closed curves.

Abstract

Any generic closed curve in the plane can be transformed into a simple closed curve by a finite sequence of local transformations called homotopy moves. We prove that simplifying a planar closed curve with $n$ self-crossings requires $\Theta(n^{3/2})$ homotopy moves in the worst case. Our algorithm improves the best previous upper bound $O(n^2)$, which is already implicit in the classical work of Steinitz; the matching lower bound follows from the construction of closed curves with large defect, a topological invariant of generic closed curves introduced by Aicardi and Arnold. Our lower bound also implies that $\Omega(n^{3/2})$ facial electrical transformations are required to reduce any plane graph with treewidth $\Omega(\sqrt{n})$ to a single vertex, matching known upper bounds for rectangular and cylindrical grid graphs. More generally, we prove that transforming one immersion of $k$ circles with at most $n$ self-crossings into another requires $\Theta(n^{3/2} + nk + k^2)$ homotopy moves in the worst case. Finally, we prove that transforming one noncontractible closed curve to another on any orientable surface requires $\Omega(n^2)$ homotopy moves in the worst case; this lower bound is tight if the curve is homotopic to a simple closed curve.