We study a new graph separation problem called Multiway Near-Separator. Given an undirected graph $G$, integer $k$, and terminal set $T \subseteq V(G)$, it asks whether there is a vertex set $S \subseteq V(G) \setminus T$ of size at most $k$ such that in graph $G-S$, no pair of distinct terminals can be connected by two pairwise internally vertex-disjoint paths. Hence each terminal pair can be separated in $G-S$ by removing at most one vertex. The problem is therefore a generalization of (Node) Multiway Cut, which asks for a vertex set for which each terminal is in a different component of $G-S$. We develop a fixed-parameter tractable algorithm for Multiway Near-Separator running in time $2^{O(k \log k)} * n^{O(1)}$. Our algorithm is based on a new pushing lemma for solutions with respect to important separators, along with two problem-specific ingredients. The first is a polynomial-time subroutine to reduce the number of terminals in the instance to a polynomial in the solution size $k$ plus the size of a given suboptimal solution. The second is a polynomial-time algorithm that, given a graph $G$ and terminal set $T \subseteq V(G)$ along with a single vertex $x \in V(G)$ that forms a multiway near-separator, computes a 14-approximation for the problem of finding a multiway near-separator not containing $x$.

翻译：我们研究了一个新的图分割问题，称为多路近分隔子。给定一个无向图$G$、整数$k$和终端集$T \subseteq V(G)$，该问题询问是否存在一个大小至多为$k$的顶点集$S \subseteq V(G) \setminus T$，使得在图$G-S$中，任意一对不同终端之间不能通过两条两两内部顶点不相交的路径相连。因此，在$G-S$中，每对终端最多可通过移除一个顶点而被分离。该问题因此是（节点）多路割的推广，后者要求一个顶点集使得每个终端位于$G-S$的不同连通分量中。我们为多路近分隔子设计了一个固定参数可解算法，运行时间为$2^{O(k \log k)} * n^{O(1)}$。我们的算法基于一个关于重要分离子解的新推进引理，以及两个问题特定的组成部分。第一个是一个多项式时间子程序，用于将实例中的终端数量缩减至解规模$k$加上给定次优解规模的多项式。第二个是一个多项式时间算法，给定图$G$、终端集$T \subseteq V(G)$以及一个构成多路近分隔子的单顶点$x \in V(G)$，该算法能计算出一个不包含$x$的多路近分隔子问题的14-近似解。