Graph separation is a central tool in parameterized algorithm design, and important separators are among its most successful ingredients. They yield small, structured families of separators that can be enumerated efficiently, and underlie fixed-parameter algorithms for many problems. However, this framework fundamentally breaks down in cut-uncut settings, where one must separate terminal sets while preserving connectivity inside specified groups of terminals. In such problems, the classical reachability-based notion of importance no longer captures the separators that matter. We introduce connectivity-preserving important separators, a new framework for cut problems with connectivity constraints. Our main result shows that this family is highly structured: the number of connectivity-preserving important separators of size at most $k$ is $2^{O(k \log k)}$, and they can be enumerated within the same bound up to polynomial factors. As an application, we obtain improved fixed-parameter algorithms for Node Multiway Cut-Uncut. In particular, when the number of equivalence classes is constant - including 2-Sets Cut-Uncut - our approach yields a $2^{O(k \log k)}$ running time, improving on the previous $2^{O(k^2 \log k)}$ dependence. More broadly, our results show that separator-based methods can be extended from pure disconnection problems to problems that simultaneously require separation and preservation of connectivity.

翻译：图分离是参数化算法设计的核心工具，而重要分离器是其最成功的组成部分之一。它们能够生成可高效枚举的小型结构化分离器族，并为许多问题提供固定参数算法。然而，这一框架在割-非割设定下根本性地失效——在此类问题中，必须在保持指定终端组内连通性的同时分离终端集。在经典基于可达性的重要性概念下，关键分离器不再能被有效捕捉。我们提出保持连通性的重要分离器，这是一种面向具有连通性约束的割问题的新框架。主要结果表明该结构具有高度组织性：大小不超过$k$的保持连通性重要分离器的数量为$2^{O(k \log k)}$，并且可在多项式因子内实现相同上界的枚举。作为应用，我们改进了节点多路割-非割问题的固定参数算法。特别地，当等价类数量为常数时（包括2-集合割-非割），我们的方法实现了$2^{O(k \log k)}$的运行时间，优于此前$2^{O(k^2 \log k)}$的依赖关系。更广泛而言，我们的结果证明基于分离器的方法可从纯断开问题扩展到同时要求分离与保持连通性的问题。