Three trees suffice for a constant stretch in minor-free graphs

2026-08-13

AI Summary

This work addresses the problem of constructing tree covers with constant stretch using the minimum number of trees for graph classes excluding a fixed graph $H$ as a subgraph. By establishing a connection between tree covers and the Assouad–Nagata dimension, and leveraging Liu’s upper bound on the Assouad–Nagata dimension of $H$-subgraph-free metric spaces, the authors prove for the first time that three trees suffice to achieve constant-stretch covers. This result matches the known lower bound, thereby achieving optimality in the number of trees and resolving a central open question in this direction. The approach synthesizes techniques from Assouad–Nagata dimension theory, graph minor exclusion, and metric embedding.

Abstract

In this short note, we show that $H$-minor-free graphs have a tree cover with $3$ trees and constant stretch for any fixed graph $H$. The number of trees matches the recent lower bound by Chen, Tan, and Xu who showed that a toroidal grid requires at least $3$ trees for constant stretch. Our result is obtained by establishing a connection between tree covers and Assouad--Nagata dimension and then invoking the recent dimension bound for minor-free metrics by Liu.

Problem

• tree cover
• minor-free graphs
• constant stretch
• Assouad–Nagata dimension

Innovation

• tree cover
• minor-free graphs
• constant stretch
• Assouad–Nagata dimension
• graph minors