The basic (and traditional) crossing number problem is to determine the minimum number of crossings in a topological drawing of an input graph in the plane. We develop a unified framework yielding fixed-parameter tractable (FPT) algorithms for many generalized crossing number problems. Our framework takes the following form. We fix a surface S and a class D of "allowed" topological drawings of graphs in S (e.g., some class of drawings with at most t crossings). We assume that testing membership in D can be done algorithmically, and that restricting a drawing in D, extending it without adding any crossing, or transforming it with a self-homeomorphism of S yields a drawing that is also in D. Then deciding whether an input graph G has a drawing in D, and computing one if it is the case, is fixed-parameter tractable in (essentially) the genus of S and the maximum number of crossings in a drawing in D. More generally, we may take as input an edge-colored graph and distinguish crossings by the colors of the involved edges; and we may allow a bounded number of edge removals and vertex splits on G before drawing it. The proof is a reduction to the embeddability of a graph on a two-dimensional simplicial complex. This implies, in a unified way, linear or quadratic FPT algorithms for many topological crossing number variants established in the graph drawing community. Some of these variants already had previously published FPT algorithms, mostly relying on Courcelle's metatheorem, but our algorithms have a better runtime. Moreover, our framework extends to these crossing number variants in any fixed surface, and also allows us to fix the rotation system of the drawing of a graph in some variants.

翻译：基本的（且传统的）交叉数问题是在平面中确定输入图的拓扑绘图的最小交叉数。我们开发了一个统一的框架，为许多广义交叉数问题提供固定参数可解（FPT）算法。该框架具有以下形式：我们固定一个曲面S和一个图在S中的“允许”拓扑绘图类D（例如，最多包含t个交叉的某类绘图）。我们假设可以算法性地判定D中的成员资格，并且限制D中的绘图、在不增加任何交叉的情况下扩展它、或通过S的自身同胚变换它，得到的绘图仍然属于D。那么，判定输入图G是否具有D中的绘图，并在存在时计算一个这样的绘图，在（本质上）曲面S的亏格和D中绘图的最大交叉数方面是固定参数可解的。更一般地，我们可以将边着色图作为输入，并根据所涉及边的颜色区分交叉；此外，我们允许在绘制G之前对其进行有界数量的边删除和顶点分裂。该证明可归约为图在二维单纯复形上的可嵌入性问题。这统一地推导出图绘制领域已建立的许多拓扑交叉数变体问题的线性或二次FPT算法。其中一些变体已有先前发表的FPT算法（主要依赖Courcelle元定理），但我们的算法具有更优的运行时间。此外，我们的框架可将这些交叉数变体推广到任意固定曲面，并在某些变体中允许固定图的绘图的旋转系统。