Graph separation problems are a cornerstone of parameterized complexity, often tackled using the "Important Separators" technique introduced by Marx. While this technique is powerful for standard separation problems, it is inapplicable to problems with connectivity constraints, where the goal is to separate terminal sets while maintaining the internal connectivity of specific components (e.g., Node Multiway Cut-Uncut and 2-sets cut-uncut). In such settings, the standard branching strategies for enumerating important separators fail, and the solution space becomes complex and seemingly unstructured. In this paper, we introduce the framework of Connectivity-Preserving (CP) Important Separators. We prove that the number of CP-important separators of size $k$ is $2^{O(k\log k)}$. Leveraging this bound, we present an efficient algorithm to enumerate all such separators. Our approach relies on a fundamental property regarding the union of minimum separators, which allows us to characterize valid separators by systematically "repairing" the connectivity violations of unconstrained separators. As a primary application, we present a new fixed-parameter tractable (FPT) algorithm for the Node Multiway Cut-Uncut (N-MWCU) problem. For the fundamental case of 2-Sets Cut-Uncut, and more generally whenever the number of equivalence classes is constant, we improve the running time from the previous best of $2^{O(k^2 \log k)}$ to $2^{O(k \log k)}$. Crucially, our approach avoids the heavy machinery of randomized contractions (and their expensive derandomization) employed by previous work, replacing it with a direct enumeration algorithm that reduces both the exponential dependence on the parameter $k$ and the polynomial overhead.

翻译：图分隔问题是参数化复杂度的基石领域，通常采用Marx提出的“重要分隔子”技术进行处理。尽管该技术在标准分隔问题中表现出强大能力，却不适用于具有连通性约束的问题——这类问题要求在分隔终端集的同时，保持特定组件内部的连通性（例如节点多路割与不割问题及2-集合割与不割问题）。在此类场景中，枚举重要分隔子的标准分支策略会失效，解空间变得复杂且看似缺乏结构。本文提出保持连通性（CP）重要分隔子的理论框架，证明规模为$k$的CP-重要分隔子数量上界为$2^{O(k\log k)}$。基于该界限，我们提出枚举所有此类分隔子的高效算法。本方法依赖于关于最小分隔子并集的基本性质，通过系统“修复”无约束分隔子的连通性违规来刻画有效分隔子。作为核心应用，我们针对节点多路割与不割（N-MWCU）问题提出新的固定参数可解（FPT）算法。对于2-集合割与不割这一基础情形（以及更一般地当等价类数量为常数时），我们将运行时间从现有最优结果$2^{O(k^2 \log k)}$改进至$2^{O(k \log k)}$。关键创新在于：本方法避免了先前研究采用的随机收缩技术（及其昂贵的去随机化过程），代之以直接枚举算法，从而同时降低参数$k$的指数依赖度与多项式开销。