A matching cut is a partition of the vertex set of a graph into two sets A and B such that each vertex has at most one neighbor in the other side of the cut. The Matching Cut problem asks whether a graph has a matching cut, and has been intensively studied in the literature. Motivated by a question posed by Komusiewicz et al. [Discrete Applied Mathematics, 2020], we introduce a natural generalization of this problem, which we call d -Cut: for a positive integer d, a d-cut is a bipartition of the vertex set of a graph into two sets A and B such that each vertex has at most d neighbors across the cut. We generalize (and in some cases, improve) a number of results for the Matching Cut problem. Namely, we begin with an NP-hardness reduction for d -Cut on \((2d+2)\)-regular graphs and a polynomial algorithm for graphs of maximum degree at most \(d+2\). The degree bound in the hardness result is unlikely to be improved, as it would disprove a long-standing conjecture in the context of internal partitions. We then give FPT algorithms for several parameters: the maximum number of edges crossing the cut, treewidth, distance to cluster, and distance to co-cluster. In particular, the treewidth algorithm improves upon the running time of the best known algorithm for Matching Cut. Our main technical contribution, building on the techniques of Komusiewicz et al. [DAM, 2020], is a polynomial kernel for d -Cut for every positive integer d, parameterized by the vertex deletion distance of the input graph to a cluster graph. We also rule out the existence of polynomial kernels when parameterizing simultaneously by the number of edges crossing the cut, the treewidth, and the maximum degree. Finally, we provide an exact exponential algorithm slightly faster than the naive brute force approach running in time \(\mathcal {O}^*\!\left( 2^n\right)\).

甲匹配切口是顶点集合的曲线图中的分区为两个集合甲和乙使得每个顶点有在切口的另一侧至多一个相邻。的匹配剪切问题询问的曲线是否具有匹配的切口，并已在文献中被广泛研究。受Komusiewicz等人提出的问题的启发。[离散应用数学，2020年]，我们引入了对这个问题的自然概括，我们称之为d -Cut：对于正整数d，d - cut是图的顶点集分成两部分的集合A和B这样每个顶点在切割处最多具有d个邻居。我们针对Matching Cut问题归纳（并在某些情况下进行改进）许多结果。即，我们从\（（2d + 2）\） -正则图上的d -Cut的NP硬度降低开始，到最大程度为\（d + 2 \）的图的多项式算法。硬度结果的约束程度不太可能得到改善，因为它将证明在内部分隔的背景下长期存在的推测。然后我们给FPT 几个参数的算法：穿过切割的最大边数，树宽，到聚类的距离以及到共聚的距离。特别地，树宽算法改善了最著名的Matching Cut算法的运行时间。我们的主要技术贡献是建立在Komusiewicz等人的技术之上。[DAM，2020]是d 的多项式内核-切每正整数d，由输入图到聚类图的顶点删除距离来参数化。当同时通过切割的边数，树宽和最大程度进行参数化时，我们还排除了多项式内核的存在。最后，我们提供了一种精确的指数算法，该算法比在时间\（\ mathcal {O} ^ * \！\ left（2 ^ n \ right）\）中运行的朴素蛮力方法要快一些。