Proper Minor-Closed Classes of Graphs have Assouad-Nagata Dimension 2

TL;DR

The paper proves that every proper minor-closed class of graphs has Assouad–Nagata dimension at most 2, strengthening prior results that used asymptotic dimension. It develops a versatile toolkit—path rerouting, colored decompositions, complete (weighted) torsos, and strong-construction frameworks on tree-decompositions—to transfer control functions and bound ANdim via genus- and treewidth-bounded components. A key contribution is the vindication that subdivision-closed classes have bounded ANdim if and only if they exclude a fixed minor, leading to a precise dichotomy that connects minor structure to metric dimension. The results have implications for embeddings, structure theory, and algorithmic approximations (e.g., TSP) by providing tight large-scale dimension bounds for broad graph families. The work integrates and extends the Graph Minor Theory program to the Assouad–Nagata setting, clarifying the landscape of dimension bounds across minor-closed and subdivision-closed classes.

Abstract

Asymptotic dimension and Assouad-Nagata dimension are measures of the large-scale shape of a class of graphs. Bonamy, Bousquet, Esperet, Groenland, Liu, Pirot, and Scott [J. Eur. Math. Society] showed that any proper minor-closed class has asymptotic dimension 2, dropping to 1 only if the treewidth is bounded. We improve this result by showing it also holds for the stricter Assouad-Nagata dimension. We also characterise when subdivision-closed classes of graphs have bounded Assouad-Nagata dimension.

Figures (1)

• Figure 1: A diagram of the separation $(G_i,\widetilde{G}_i)$ of $G$ for some $i\in \{1,\dots,a\}$, with several relevant sets of vertices near the separator $S_i$ labelled. Vertices and edges of $G$ are not depicted, instead the dotted line denotes a region in which all vertices and edges are contained. Observe that any $r$-path $P$ from a vertex in $V(G_i)\setminus N_i^2$ to a vertex in $V(\widetilde{G}_i)\cup N_i^2$ must contain a vertex in both $S_i^2$ and $S_i^3$, which are monochromatic of different colours, depicted here as red and blue. Thus, $P$ cannot be monochromatic. Additionally, observe that any $r$-path from a vertex in $S_i^2$ to a vertex in $\widetilde{G}_i\cup N_i^1$ must contain a vertex in $S_i^1$.

Theorems & Definitions (50)

• Theorem 1: Bonamy
• Lemma 2
• Theorem 3
• Theorem 4
• Proposition 4
• Proposition 4
• Proposition 5
• proof
• Corollary 6
• proof