Let $G$ be an undirected graph, and $s,t$ distinguished vertices of $G$. A minimal $s,t$-separator is an inclusion-wise minimal vertex-set whose removal places $s$ and $t$ in distinct connected components. We present an algorithm for listing the minimal $s,t$-separators of a graph in non-decreasing order of cardinality, in polynomial-delay. This problem finds applications in various algorithms parameterized by treewidth, which include query evaluation in relational databases, probabilistic inference, and many more. In the process, we prove several results that are of independent interest. We establish a new island of tractability to the intensively studied 2-disjoint connected subgraphs problem, which is NP-complete even for restricted graph classes that include planar graphs, and prove new characterizations of minimal $s,t$-separators. Ours is the first to present a ranked enumeration algorithm for minimal separators where the delay is polynomial in the size of the input graph.

翻译：设$G$为一个无向图，$s,t$为其两个指定顶点。极小$s,t$-分离子是一个按包含关系极小的顶点集，其移除使得$s$和$t$位于不同的连通分量中。我们提出一种算法，能够以多项式延迟按基数非递减顺序列出图中所有极小$s,t$-分离子。该问题在以树宽为参数的各种算法中具有应用价值，包括关系数据库中的查询评估、概率推理等众多领域。在研究过程中，我们证明了一些具有独立意义的结果。针对被广泛研究的2-不相交连通子图问题（即使对于包含平面图在内的受限图类，该问题也是NP完全的），我们建立了新的可解性理论；并提出了极小$s,t$-分离子的新刻画。这是首个以输入图规模多项式延迟对极小分离子进行排序枚举的算法。