We confirm a conjecture of Gartland and Lokshtanov [arXiv:2007.08761]: if for a hereditary graph class $\mathcal{G}$ there exists a constant $k$ such that no member of $\mathcal{G}$ contains a $k$-creature as an induced subgraph or a $k$-skinny-ladder as an induced minor, then there exists a polynomial $p$ such that every $G \in \mathcal{G}$ contains at most $p(|V(G)|)$ minimal separators. By a result of Fomin, Todinca, and Villanger [SIAM J. Comput. 2015] the latter 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 $\mathcal{G}$. Furthermore, as shown by Gartland and Lokshtanov, our result implies a full dichotomy of hereditary graph classes defined by a finite set of forbidden induced subgraphs into tame (admitting a polynomial bound of the number of minimal separators) and feral (containing infinitely many graphs with exponential number of minimal separators).

翻译：我们确认Gartland 和 Lokshtanov [arXiv: 2007.08761] 的猜想:如果对于遗传图形类来说, $\ mathcal{G} 美元存在一个恒定的美元美元美元, 以至于$\ mathcal{G} 没有一个成员以诱导的子集法或 $k$-kunny-ladder 以诱导的未成年人的形式含有一个k$- creaty, 那么就存在一个多数值的美元和Lokshtanov [ar: 2007. 08761] : 如果对于遗传图形类来说, $\ mathcal{G} $( $V) $( G) $( ) $( $) $( $+V ( G) ) $( ) $( ) $( ) $( $( $) $( $)( $) $( $) $( ) $( ) $ ( $) $( $ ( $) $ ( $) $ ( ) $ ( ) $ ( $) $ ( ) ) $( $ ( $ ( $) $) listpar ( $) ) ) ) list) list g) 最小的计算, listpar ( g) 最小的计算 seq) separ ( g) $( $( g) $( ) ) $( g) $( g) $( ) ) $( g) $( g) $( g) ) ) $( ) ) ) ) ) ) ) $( ) $( ) ) $( $( $( ) $( ) ) ) $( $( ) ) ) $( ) $( ) ) ) $( ) ) $( $( ) ) $( ) ) ) $( $( $( $( $(