A graph class is tame if it admits a polynomial bound on the number of minimal separators, and feral if it contains infinitely many graphs with exponential number of minimal separators. The former entails the existence of polynomial-time algorithms for Maximum Weight Independent Set, Feedback Vertex Set, and many other problems, when restricted to an input graph from a tame class, by a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015]. 
In the talk, we show a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame and feral. To obtain the full dichotomy, we confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class C there exists a constant k such that no member of C contains a k-creature or a k-skinny-ladder as an induced minor, then there exists a polynomial p such that every graph G from C contains at most p(|V(G)|) minimal separators. 
Joint work with Jakub Gajarský, Lars Jafke, Paloma T. Lima, Marcin Pilipczuk, Paweł Rzążewski, and Uéverton S. Souza.