The basic crossing number problem is to determine the minimum number of crossings in a topological drawing of an input graph in the plane. In this paper, we develop fixed-parameter tractable (FPT) algorithms for various generalized crossing number problems in the plane or on surfaces. Our first result is on the color-constrained crossing problem, in which edges of the input graph G are colored, and one looks for a drawing of G in the plane or on a given surface in which the total number of crossings involving edges of colors i and j does not exceed a given upper bound Mij. We give an algorithm for this problem that is FPT in the total number of crossings allowed and the genus of the surface. It readily implies an FPT algorithm for the joint crossing number problem. We also give new FPT algorithms for several other graph drawing problems, such as the skewness, the edge crossing number, the splitting number, the gap-planar crossing number, and their generalizations to surfaces. Our algorithms are reductions to the embeddability of a graph on a two-dimensional simplicial complex, which admits an FPT algorithm by a result of Colin de Verdi\`ere and Magnard [ESA 2021].

翻译：基础交叉数问题旨在确定输入图在平面拓扑绘制中的最小交叉次数。本文针对平面或曲面上的各类广义交叉数问题，发展了固定参数可处理（FPT）算法。我们的首个成果关注颜色约束交叉问题：输入图G的边被着色，需在平面或给定曲面上寻找G的一种绘制，使得颜色i与j边之间的交叉总数不超过给定上界Mij。针对该问题，我们提出一种算法，其FPT复杂度取决于允许的交叉总数与曲面亏格。该算法可直接推导出联合交叉数问题的FPT算法。我们还为若干其他图绘制问题提供了新的FPT算法，包括偏斜度、边交叉数、分裂数、间隙平面交叉数及其在曲面上的推广。我们的算法通过归约至图在二维单纯复形上的可嵌入性实现，而根据Colin de Verdière与Magnard的研究成果[ESA 2021]，该问题存在FPT算法。