论文标题
没有大生物和瘦梯子的驯服图
Taming graphs with no large creatures and skinny ladders
论文作者
论文摘要
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多项式$ p $，以使每个$ g \ in \ mathcal {g} $最多包含$ p（| v（g）|）$最小分隔符。由于Fomin，Todinca和Villanger的结果[Siam J. Comput。 [2015]后者需要以最大重量独立集，反馈顶点集和许多其他问题的多项式时间算法的存在，并且仅限于$ \ Mathcal {g} $的输入图。此外，如Gartland和Lokshtanov所示，我们的结果暗示了遗传图类别的完整二分法，这是由一组有限的禁止的诱导子图（承认最小分隔符数量的多项式限制）定义的，并包含具有数量的许多图形，具有数量的数量数量，具有指数分隔符的数量）。
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).
