Self-overlapping Curves Revisited

TL;DR

This paper explores the computational complexity of recognizing when self-overlapping curves are projections of embedded surfaces in space, revealing NP-completeness in some cases and efficient algorithms when additional casing information is provided.

Contribution

It demonstrates NP-completeness for certain surface recognition problems and provides a linear-time algorithm for recognizing cased curves as projections of embedded surfaces.

Findings

Recognizing if an immersed disk is a projection of an embedded surface is NP-complete.

With casing information, the recognition problem is solvable in linear time.

A surface with a single boundary crossing n times has at most 2^{n/2} distinct spatial embeddings.

Abstract

A surface embedded in space, in such a way that each point has a neighborhood within which the surface is a terrain, projects to an immersed surface in the plane, the boundary of which is a self-intersecting curve. Under what circumstances can we reverse these mappings algorithmically? Shor and van Wyk considered one such problem, determining whether a curve is the boundary of an immersed disk; they showed that the self-overlapping curves defined in this way can be recognized in polynomial time. We show that several related problems are more difficult: it is NP-complete to determine whether an immersed disk is the projection of a surface embedded in space, or whether a curve is the boundary of an immersed surface in the plane that is not constrained to be a disk. However, when a casing is supplied with a self-intersecting curve, describing which component of the curve lies above and which below at each crossing, we may determine in time linear in the number of crossings whether the cased curve forms the projected boundary of a surface in space. As a related result, we show that an immersed surface with a single boundary curve that crosses itself n times has at most 2^{n/2} combinatorially distinct spatial embeddings, and we discuss the existence of fixed-parameter tractable algorithms for related problems.