Graphs Excluding a Minor in Blowups of Treewidth 3 Graphs

TL;DR

The paper tackles the problem of improving the treewidth blowup bound for $K_h$-minor-free graphs from treewidth-4 to treewidth-3 via a refined blowup framework. Building on the Graph Minor Structure Theorem and recent almost-embedding techniques, it develops a two-stage strategy: first almost-partitioning graphs that admit $K_h$-minor-free almost-embeddings, and then partitioning using a Raise-based consolidation to control width and loss. A central technical feature is the decomposability parameter, which ensures torsos admit good almost-partitions and remains stable under sub-decompositions; this enables reattachment of components with bounded growth of width and controlled loss. The paper also provides algorithmic tools, including reductions to plane+quasi-vortex embeddings, layerings, and a mechanism to transform complex almost-embeddings into near-planar forms by means of cutting subgraphs and adjustments. Overall, the work advances the structural understanding of minor-free graphs and yields a near-optimal blowup decomposition, with potential pathways to further tighten the treewidth bound toward the theoretical minimum.

Abstract

Alon, Seymour, and Thomas [J. Amer. Math. Soc. 1990] famously showed that every $n$-vertex $K_h$-minor-free graph has treewidth $O_h(\sqrt{n})$. Recently, Distel, Dujmović, Eppstein, Hickingbotham, Joret, Micek, Morin, Seweryn, and Wood [SIAM J. Discrete Math. 2024] refined this by showing that these graphs are $O_h(\sqrt{n})$-blowups of treewidth $4$ graphs. We improve this by showing that these graphs are $O_h(\sqrt{n})$-blowups of treewidth $3$ graphs.

Theorems & Definitions (199)

• Theorem 1.0.1
• Theorem 1.0.2
• Theorem 1.0.3
• proof
• Theorem 1.3.1
• Theorem 1.3.2
• Theorem 1.3.3
• proof
• Theorem 1.4.2
• Theorem 1.4.3