A minimal separator in a graph is an inclusion-minimal set of vertices that separates some fixed pair of nonadjacent vertices. A graph class is said to be tame if there exists a polynomial upper bound for the number of minimal separators of every graph in the class, and feral if it contains arbitrarily large graphs with exponentially many minimal separators. Building on recent works of Gartland and Lokshtanov [SODA 2023] and Gajarsk\'y, Jaffke, Lima, Novotn\'a, Pilipczuk, Rz\k{a}\.zewski, and Souza [arXiv, 2022], we show that every graph class defined by a single forbidden induced minor or induced topological minor is either tame or feral, and classify the two cases. This leads to new graph classes in which Maximum Weight Independent Set and many other problems are solvable in polynomial time. We complement the classification results with polynomial-time recognition algorithms for the maximal tame graph classes appearing in the obtained classifications.

翻译：在图论中，最小分隔集是指一个顶点集合，它分离某个固定的非相邻顶点对，且在该性质下是包含关系最小的。若某个图类中每个图的最小分隔集数量均存在多项式上界，则称该类为驯服的；若该类包含具有指数级多个最小分隔集的任意大图，则称其为野生的。基于Gartland与Lokshtanov [SODA 2023] 以及Gajarský、Jaffke、Lima、Novotná、Pilipczuk、Rzążewski和Souza [arXiv, 2022] 的最新研究，我们证明了每个由单一禁止诱导子式或诱导拓扑子式定义的图类要么是驯服的，要么是野生的，并对这两种情况进行了分类。这一结论导出了多个新的图类，其中最大权重独立集及许多其他问题可在多项式时间内求解。我们进一步为分类结果中出现的极大驯服图类提供了多项式时间识别算法，作为对分类结果的补充。