再论自重叠曲线
Self-overlapping Curves Revisited
摘要：
一个嵌入空间的表面，以这样一种方式，每个点都有一个邻域，在该邻域内该表面是一个地形，投影到平面中的一个浸入表面，其边界是一条自相交曲线。在什么情况下我们可以通过算法来反转这些映射？Shor和van Wyk考虑了这样一个问题，确定一条曲线是否是浸没盘的边界;他们表明，以这种方式定义的自重叠曲线可以在多项式时间内识别。我们发现，几个相关的问题是更困难的：它是NP-完全的，以确定是否沉浸盘是嵌入在空间中的表面的投影，或者是否一条曲线是沉浸表面的边界在不受约束的平面是一个磁盘。然而，当一个外壳提供了一个自相交曲线，描述曲线的哪些组成部分位于上方和下方的每个交叉点，我们可以确定在时间上线性的交叉点的数量是否套管曲线形成的投影边界的表面在空间中。作为一个相关的结果，我们表明，一个浸入表面与一个单一的边界曲线，交叉本身n倍，有最多2^{n/2}组合不同的空间嵌入，我们讨论了存在的固定参数听话的算法相关的问题。
英文摘要：
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.
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