Computing the crossing number of a graph is one of the most classical problems in computational geometry. Both it and numerous variations of the problem have been studied, and overcoming their frequent computational difficulty is an active area of research. Particularly recently, there has been increased effort to show and understand the parameterized tractability of various crossing number variants. While many results in this direction use a similar approach, a general framework remains elusive. We suggest such a framework that generalizes important previous results, and can even be used to show the tractability of deciding crossing number variants for which this was stated as an open problem in previous literature. Our framework targets variants that prescribe a partial predrawing and some kind of topological restrictions on crossings. Additionally, to provide evidence for the non-generalizability of previous approaches for the partially crossing number problem to allow for geometric restrictions, we show a new more constrained hardness result for partially predrawn rectilinear crossing number. In particular, we show W-hardness of deciding Straight-Line Planarity Extension parameterized by the number of missing edges.

翻译：计算图的交叉数是计算几何学中最经典的问题之一。该问题及其众多变体已被广泛研究，克服其普遍存在的计算困难是当前活跃的研究领域。特别是在近期，研究者们愈发致力于证明和理解各类交叉数变体的参数化可处理性。尽管该方向的许多成果采用了相似的研究思路，但通用的理论框架仍尚未建立。本文提出了一种能够推广先前重要结果的通用框架，该框架甚至可用于证明某些交叉数变体判定问题的可处理性——这些变体在以往文献中被列为开放性问题。本框架主要针对那些规定了部分预绘制及交叉点拓扑约束的变体。此外，为证明现有方法在允许几何约束的部分交叉数问题上缺乏普适性，我们针对部分预绘制的直线交叉数问题提出了新的更强约束条件下的困难性结果。具体而言，我们证明了以缺失边数为参数的直线平面性延拓判定问题具有W-困难性。