Connectivity-Preserving Important Separators: A Framework for Cut-Uncut Problems

Abstract

Important separators reduce the search space in parameterized graph separation to a small family that can be enumerated efficiently. We extend this approach to cut-uncut problems, which require separating some vertices while keeping a prescribed terminal set connected. We introduce connectivity-preserving important separators in undirected graphs. These are inclusion-minimal $s,t$-vertex separators that keep the prescribed terminals connected to $s$ and admit no such separator of no greater size whose source component is properly contained in their own. We show that there are at most $2^{O(k^2\log k)}$ such separators of size at most $k$ and give an algorithm that enumerates them in $O(2^{O(k^2\log k)}nT(n,m))$ time. Here $n$ and $m$ denote the numbers of vertices and edges, and $T(n,m)$ is the time to compute a minimum-cardinality $s,t$-separator. We apply this result to Mixed Multiway Cut-Uncut with any fixed number of terminal classes, improving the published $2^{O(p^3\log p)}$ parameter dependence to $2^{O(p^2\log p)}$, where $p$ is the sum of the vertex and edge deletion budgets. For Annotated Odd Cycle Transversal with the additional requirement that an unbounded set of undeletable terminals remain connected, we give a $2^{O(k^2\log k)}(n+m+1{)}^{O(1)}$-time algorithm. This improves the doubly exponential dependence of a direct treewidth-reduction approach. We also obtain a fixed-parameter tractable algorithm for finding a minimum-cardinality inclusion-minimal $s,t$-separator whose source component contains a prescribed set $A$ and avoids a prescribed set $B$.