Dominated Minimal Separators are Tame (Nearly All Others are Feral)
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A class F of graphs is called tame if there exists a constant k so that every graph in F on n vertices contains at most O(n^k) minimal separators, strongly-quasi-tame if every graph in F on n vertices contains at most O(n^k log n) minimal separators, and feral if there exists a constant c > 1 so that 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-creature. In a recent manuscript [Abrishami et al., Arxiv 2020] conjecture that every hereditary class 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 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. (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]. (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]. (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.
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