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arXiv · 2205.01191

Taming graphs with no large creatures and skinny ladders

Jakub Gajarský·Lars Jaffke·Paloma T. Lima·Jana Novotná·Marcin Pilipczuk·Paweł Rzążewski·Uéverton S. Souza
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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).

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Jakub Gajarský, Lars Jaffke, Paloma T. Lima, Jana Novotná, Marcin Pilipczuk, Paweł Rzążewski, Uéverton S. Souza. 2022-05-02. Taming graphs with no large creatures and skinny ladders. https://arxiv.org/abs/2205.01191

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