支配最小分隔符是驯服的（几乎所有其他分隔符都是野性的）
Dominated Minimal Separators are Tame (Nearly All Others are Feral)
摘要：
一个图类${\cal F}$称为{ \em tame}，如果存在一个常数$k$，使得${\cal F}$中的每个图在$n$顶点上至多包含$O(n^k)$最小分隔符，如果${\cal F}$中的每个图在$n$顶点上至多包含$O(n^{k \log n})$最小分隔符，和{ \em feral}，如果存在一个常数$c > 1$，使得对于任意大的$n$，${\cal F}$包含至少具有$c^n$最小分隔符的$n$ -顶点图。图类的分类驯服或野生有许多算法的后果，最近得到了相当大的关注。在寻求这种分类的过程中，一个关键的图论对象是$k$ -{ \em creature}的概念。在最近的手稿[Abrishami等人，Arxiv 2020]猜想，对于某个固定常数$k$，排除$k$ -生物的每个遗传类${\cal F}$是驯服的。我们给出了这个猜想的一个反例，并证明了一个较弱的结果：遗传类${\cal F}$是强拟驯服的，如果它对某个固定的常数$k$不包含$k$ -生物，并且每个极小分离子可以被另一个固定的常数$k'$顶点数支配.开发的工具还导致了一些独立的利益的额外结果。{ \bf（i）我们得到了由有限个禁止导出子图所定义的所有遗传图类为强拟驯服或野性的完全分类。这概括了Milani和Piva [WG'19]。{ \bf（ii）}我们证明了遗传类，排除$k$ -生物，并额外排除所有周期的长度至少$c$，对于一些常数$c$，是驯服的。这推广了[Chudnovsky等人，Arxiv 2019]。{ \bf（iii）}我们证明了每个遗传类，排除$k$ -生物，并另外排除一个完全图上的$c$顶点为一些固定的常数$c$是驯服的。
英文摘要：
A class ${\cal F}$ of graphs is called {\em tame} if there exists a constant $k$ so that every graph in ${\cal F}$ on $n$ vertices contains at most $O(n^k)$ minimal separators, {\em strongly-quasi-tame} if every graph in ${\cal F}$ on $n$ vertices contains at most $O(n^{k \log n})$ minimal separators, and {\em feral} if there exists a constant $c > 1$ so that ${\cal F}$ contains $n$-vertex graphs with at least $c^n$ minimal separators for arbitrarily large $n$. The classification of graph classes into tame or feral has numerous algorithmic consequences, and has recently received considerable attention. A key graph-theoretic object in the quest for such a classification is the notion of a $k$-{\em creature}. In a recent manuscript [Abrishami et al., Arxiv 2020] conjecture that every hereditary class ${\cal F}$ that excludes $k$-creatures for some fixed constant $k$ is tame. We give a counterexample to this conjecture and prove the weaker result that a hereditary class ${\cal F}$ is strongly quasi-tame if it excludes $k$-creatures for some fixed constant $k$ and additionally every minimal separator can be dominated by another fixed constant $k'$ number of vertices. The tools developed also lead to a number of additional results of independent interest. {\bf (i) We obtain a complete classification of all hereditary graph classes defined by a finite set of forbidden induced subgraphs into strongly quasi-tame or feral. This generalizes Milanič and Pivač [WG'19]. {\bf (ii)} We show that hereditary class that excludes $k$-creatures and additionally excludes all cycles of length at least $c$, for some constant $c$, are tame. This generalizes the result of [Chudnovsky et al., Arxiv 2019]. {\bf (iii)} We show that every hereditary class that excludes $k$-creatures and additionally excludes a complete graph on $c$ vertices for some fixed constant $c$ is tame.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
