A tame vs. feral dichotomy for graph classes excluding an induced minor or induced topological minor

TL;DR

This paper classifies graph classes excluding certain minors as either tame or feral based on the number of minimal separators, leading to new polynomial-time solvable problems and recognition algorithms for these classes.

Contribution

It provides a dichotomy classification for graph classes excluding an induced minor or topological minor, identifying tame and feral classes and their properties.

Findings

Every such class is either tame or feral.

Polynomial-time algorithms for recognizing tame classes.

Polynomial-time solvability of problems like Maximum Weight Independent Set in tame classes.

Abstract

A minimal separator in a graph is an inclusion-minimal set of vertices that separates some fixed pair of nonadjacent vertices. A graph class is said to be tame if there exists a polynomial upper bound for the number of minimal separators of every graph in the class, and feral if it contains arbitrarily large graphs with exponentially many minimal separators. Building on recent works of Gartland and Lokshtanov [SODA 2023] and Gajarsk\'y, Jaffke, Lima, Novotn\'a, Pilipczuk, Rz\k{a}\.zewski, and Souza [arXiv, 2022], we show that every graph class defined by a single forbidden induced minor or induced topological minor is either tame or feral, and classify the two cases. This leads to new graph classes in which Maximum Weight Independent Set and many other problems are solvable in polynomial time. We complement the classification results with polynomial-time recognition algorithms for the maximal tame graph classes appearing in the obtained classifications.