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

TL;DR

We address the problem of classifying graph classes defined by excluding a single induced minor or induced topological minor as tame or feral, proving that every such class falls into one of the two categories and providing a complete dichotomy. The main approach combines a tameness criterion from prior work with structural analyses of thin walks, short prisms, and short thetas, yielding exact characterizations in terms of forbidden patterns such as the diamond, butterfly, house, and 2P2. As a consequence, Maximum Weight Independent Set and related problems become polynomial-time solvable in the tame cases, and the maximal tame classes are recognizable in polynomial time. The results advance the understanding of how induced-minor and induced-topological-minor exclusions govern computational tractability, with practical implications for MWIS and related fixed-parameter and algebraic approaches.

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 Gajarský, Jaffke, Lima, Novotná, Pilipczuk, Rzążewski, 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.

Figures (5)

• Figure 1: From left to right: the $2P_2$, the diamond, the butterfly, and the house.
• Figure 2: Schematic representations of the graphs $k$-theta, $k$-prism, $k$-creature, and $k$-skinny ladder. Dotted edges may or may not exist.
• Figure 3: A schematic representation of graphs in $\mathcal{S}$, $\mathcal{T}$, and $\mathcal{M}$. Dashed edges represent paths of positive length. Dotted edges may be present or not. The figure is adapted from dallard2024detecting.
• Figure 4: The graphs $\Gamma_{2,4,4}$ (left) and $\Gamma_{2,2,3}$ (right).
• Figure 5: Some small graphs.

Theorems & Definitions (53)

• Theorem 1.1: Fomin, Todinca, and Villanger MR3311877
• Theorem 1.7
• Theorem 1.8
• Theorem 1.9
• Theorem 1.10
• Corollary 1.11
• Lemma 3.1
• proof
• Proposition 3.2
• proof