Blow-up structure of graphs excluding a tree or an apex-tree as a minor

TL;DR

This paper establishes new structural theorems for graphs excluding certain trees or apex-trees as minors, showing they can be embedded into products of graphs with bounded pathwidth or treewidth, refining previous bounds.

Contribution

It provides improved bounds on the structure of minor-excluding graphs, specifically relating to their containment in graph products with bounded parameters, advancing prior results.

Findings

Graphs excluding a tree as a minor are contained in a product of a graph with bounded pathwidth and a complete graph.

Graphs excluding an apex-tree as a minor are contained in a product of a graph with bounded treewidth and a large complete graph.

Bounds on the parameters are tight up to constant factors, improving previous exponential bounds.

Abstract

We prove blow-up structure theorems for graphs excluding a tree or an apex-tree as a minor. First, we show that for every $t$-vertex tree $T$ with $t\geq 3$ and radius $h$, and every graph $G$ excluding $T$ as a minor, there exists a graph $H$ with pathwidth at most $2h-1$ such that $G$ is contained in $H\boxtimes K_{t-2}$ as a subgraph. This improves on a recent theorem of Dujmovi\'c, Hickingbotham, Joret, Micek, Morin, and Wood (2024), who proved the same result but with a larger bound on the order of the complete graph in the product. Second, we show that for every $t$-vertex tree $T$ with $t\geq 2$, radius $h$ and maximum degree $d$, and every graph $G$ excluding the apex-tree $T^+$ as a minor, where $T^+$ is the tree obtained by adding a universal vertex to $T$, there exists a graph $H$ with treewidth at most $4h-1$ such that $G$ is contained in $H\boxtimes K_{2(t-1)d}$. The bound on the treewidth of $H$ is best possible up to a factor $2$, and improves on a $2^{h+2}-4$ bound that follows from a recent result of Dujmovi\'c, Hickingbotham, Hodor, Joret, La, Micek, Morin, Rambaud, and Wood (2024).