We propose two fixed-parameter tractable algorithms for the weighted Max-Cut problem on embedded 1-planar graphs parameterized by the crossing number k of the given embedding. A graph is called 1-planar if it can be drawn in the plane with at most one crossing per edge. Our algorithms recursively reduce a 1-planar graph to at most $3^k$ planar graphs, using edge removal and node contraction. Our main algorithm then solves the Max-Cut problem for the planar graphs using the FCE-MaxCut introduced by Liers and Pardella [23]. In the case of non-negative edge weights, we suggest a variant that allows to solve the planar instances with any planar Max-Cut algorithm. We show that a maximum cut in the given 1-planar graph can be derived from the solutions for the planar graphs. Our algorithms compute a maximum cut in an embedded weighted 1-planar graph with n nodes and k edge crossings in time $\mathcal{O}(3^k\cdot n^{3/2}\log n)$.

针对由给定嵌入的交叉数k参数化的嵌入式1-平面图的加权Max-Cut问题，我们提出了两种固定参数易处理算法。如果可以在平面中以每条边最多一个交叉的方式绘制图形，则将其称为1-平面。我们的算法将1平面图递归减少到最大$3^{ķ}$平面图，使用边缘去除和节点收缩。然后，我们的主要算法使用Liers和Pardella [23]引入的FCE-MaxCut解决平面图的Max-Cut问题。在非负边缘权重的情况下，我们建议使用一种变体，该变体允许使用任何平面Max-Cut算法求解平面实例。我们表明，可以从平面图的解中得出给定的1平面图的最大割宽。我们的算法在n个节点和k个边缘交叉点的时间的嵌入式加权1平面图中计算最大割点$\mathcal{Ø}（3^{ķ}\cdot {ñ}^{3/2}\mathrm{日志}ñ）$。