Using the concepts of Eulerian-spanning set and coboundary operator, we generalize Hadlock's conversion of the maxcut problem on planar graphs to one on general graphs with non-negative weights. Using our conversion, we can explore algorithms for maxcut beyond the class of planar graphs. We obtain a Fixed-Parameter Tractable algorithm for $k$-contraction apex graphs. Specifically, our algorithm can be applied to graphs with crossing number $k$, giving an $O(2^k(n+k)^{3/2}\log (n+k))$-time algorithm that matches the best known results when restricted to non-negative weights.

翻译：利用欧拉生成集与上边界算子的概念，我们将Hadlock提出的平面图最大割问题转化方法推广至具有非负权重的一般图。通过这种转化，我们得以探索平面图类之外的最大割算法。针对$k$-收缩顶图，我们获得了一种固定参数可解算法。具体而言，该算法可应用于交叉数为$k$的图，其时间复杂度为$O(2^k(n+k)^{3/2}\log (n+k))$，在权重非负的约束条件下与已知最优结果相匹配。