Taming graphs with no large creatures and skinny ladders

TL;DR

This paper proves that hereditary graph classes excluding large creatures and skinny ladders have polynomially bounded minimal separators, enabling efficient algorithms for various problems and establishing a dichotomy in class complexity.

Contribution

It confirms a conjecture linking forbidden induced subgraphs to polynomial bounds on minimal separators, leading to algorithmic and structural insights.

Findings

Hereditary classes with no large creatures or skinny ladders have polynomially bounded minimal separators.

This result enables polynomial-time algorithms for problems like Maximum Weight Independent Set.

It establishes a dichotomy between tame and feral hereditary classes based on forbidden subgraphs.

Abstract

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).