Dominated Minimal Separators are Tame (Nearly All Others are Feral)

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 Milani\v{c} and Piva\v{c} [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.