Connectivity-Preserving Important Separators: Enumeration and an Improved FPT Algorithm for Node Multiway Cut-Uncut

We develop a framework for handling graph separation problems with connectivity constraints. Extending the classical concept of important separators, we introduce and analyze connectivity-preserving important separators, which are important separators that not only disconnect designated terminal sets $A$ and $B$ but also satisfy an arbitrary set of connectivity constraints over the terminals. These constraints can express requirements such as preserving the internal connectivity of each terminal set, enforcing pairwise connections defined by an equivalence relation, or maintaining reachability from a specified subset of vertices. We prove that for any graph $G=(V,E)$, terminal sets $A,B\subseteq V$, and integer $k$, the number of important $A,B$-separators of size at most $k$ satisfying a set of connectivity constraints is bounded by $2^{O(k\log k)}$, and that all such separators can be enumerated within $O(2^{O(k\log k)} \cdot n \cdot T(n,m))$ time, where $T(n,m)$ is the time required to compute a minimum $s,t$-separator. As an application, we obtain a new fixed-parameter-tractable algorithm for the Node Multiway Cut-Uncut (N-MWCU) problem, parameterized by $k$, the size of the separator set. The algorithm runs in $O(2^{O(k\log k)} \cdot n \cdot m^{1+o(1)})$ time for graphs with polynomially-bounded integer weights. This significantly improves the dependence on $k$ from the previous $2^{O(k^2\log k)}$ to $2^{O(k\log k)}$, thereby breaking a long-standing barrier, and simultaneously improves the polynomial factors. Our framework generalises the important-separator paradigm to separation problems in which the deletion set must satisfy both cut and uncut constraints on terminal subsets, thus offering a refined combinatorial foundation for designing fixed-parameter algorithms for cut-uncut problems in graphs.