The Excluded Tree Minor Theorem Revisited

TL;DR

This paper revisits the Excluded Tree Minor Theorem by showing that for any tree $T$ of radius $h$, there exists $c$ such that every $T$-minor-free graph $G$ is contained in $H\boxtimes K_c$ for some graph $H$ with $\mathrm{pw}(H)\le 2h-1$, strengthening Robertson–Seymour's result. The authors leverage graph product structure and $H$-partitions, establishing the equivalence that $G\subseteq H\boxtimes K_p$ if and only if $G$ has an $H$-partition of width at most $p$, and prove a quantitative bound: for $T$ with $t$ vertices, radius $h$, and maximum degree $d$, every $T$-minor-free $G$ is contained in $H\boxtimes K_{(d+h-2)(t-1)}$ with $\mathrm{pw}(H)\le 2h-1$. They also derive tight lower bounds, showing $g(T)\in[h-1,2h-1]$, with the upper bound realized by complete ternary trees and the lower bound by paths, illustrating that the radius controls the structural complexity. Together, the results enrich the graph product structure theory for minor-free graphs and demonstrate that radius is the natural parameter governing excluded-tree minor phenomena.

Abstract

We prove that for every tree $T$ of radius $h$, there is an integer $c$ such that every $T$-minor-free graph is contained in $H\boxtimes K_c$ for some graph $H$ with pathwidth at most $2h-1$. This is a qualitative strengthening of the Excluded Tree Minor Theorem of Robertson and Seymour (GM I). We show that radius is the right parameter to consider in this setting, and $2h-1$ is the best possible bound.

Theorems & Definitions (12)

• Theorem 1
• Lemma 2: UTW
• Proposition 2
• Proposition 3
• Lemma 4
• Theorem 6
• Lemma 7
• proof
• Lemma 8
• proof