Enumerating Minimal Separators in Ranked Order

Let $G$ be an $n$-vertex graph, and $s,t$ vertices of $G$. We present an efficient algorithm which enumerates the set of minimal $st$-separators of $G$ in ascending order of cardinality, with a delay of $O(n^{3.5})$ per separator. In particular, we present an algorithm that lists, in ascending order of cardinality, all minimal separators with at most $k$ vertices. In that case, we show that the delay of the enumeration algorithm is $O(kn^{2.5})$ per separator. Our process is based on a new method that can decide, in polynomial time, whether the set of minimal separators under certain inclusion, exclusion, and cardinality constraints is empty.