Given a positive integer $d$, the d-CUT is the problem of deciding if an undirected graph $G=(V,E)$ has a cut $(A,B)$ such that every vertex in $A$ (resp. $B$) has at most $d$ neighbors in $B$ (resp. $A$). For $d=1$, the problem is referred to as MATCHING CUT. Gomes and Sau, in IPEC 2019, gave the first fixed parameter tractable algorithm for d-CUT parameterized by maximum number of the crossing edges in the cut (i.e. the size of edge cut). However, their paper doesn't provide an explicit bound on the running time, as it indirectly relies on a MSOL formulation and Courcelle's Theorem. Motivated by this, we design and present an FPT algorithm for d-CUT for general graphs with running time $2^{O(k\log k)}n^{O(1)}$ where $k$ is the maximum size of the edge cut. This is the first FPT algorithm for the d-CUT and MATCHING CUT with an explicit dependence on this parameter. We also observe that there is no algorithm solving MATCHING CUT in time $2^{o(k)}n^{O(1)}$ where $k$ is the maximum size of the edge cut unless ETH fails.

翻译：给定正整数$d$，d-CUT问题旨在判定一个无向图$G=(V,E)$是否存在割$(A,B)$，使得$A$（相应地，$B$）中的每个顶点在$B$（相应地，$A$）中至多有$d$个邻居。当$d=1$时，该问题被称为匹配割问题。Gomes与Sau在IPEC 2019中首次提出了针对d-CUT的固定参数可解算法，其参数为割中交叉边的最大数量（即边割的规模）。然而，该论文未给出运行时间的显式上界，因其间接依赖于MSOL公式表述与Courcelle定理。受此启发，我们针对一般图设计并提出了d-CUT的固定参数可解算法，其运行时间为$2^{O(k\log k)}n^{O(1)}$，其中$k$为边割的最大规模。这是首个针对d-CUT及匹配割问题、且对该参数具有显式依赖关系的固定参数可解算法。我们还指出：除非指数时间假设不成立，否则不存在以$2^{o(k)}n^{O(1)}$时间求解匹配割问题的算法，其中$k$为边割的最大规模。