Basis Number of Graphs Excluding Minors

TL;DR

The paper proves that graphs excluding a fixed minor $H$ have bounded basis number $ ext{bn}(G)$, addressing a central question about minor-closed graph classes. It develops a two-pronged approach: first establishing that graphs with bounded treewidth (and more generally certain tree-decompositions) have bounded bn, and then using the Graph Minor Structure Theorem to assemble $H$-minor-free graphs from bricks that are almost embeddable in surfaces (with apices and vortices). The authors combine tree-decomposition techniques, Simon’s Factorisation Forests, and hypergraph substitutions to control basis-number growth across adhesions, culminating in a double-exponential bound that can be improved to a polynomial bound by independent works. The results connect basis-number theory to the broader theory of graph minors, showing that monotone classes with bounded bn correspond precisely to those excluding a minor. The work has implications for understanding cycle-space generation and could influence related structural graph theory questions and potential algorithmic applications where cycle bases are central.

Abstract

The basis number of a graph $G$ is the minimum $k$ such that the cycle space of $G$ is generated by a family of cycles using each edge at most $k$ times. A classical result of Mac Lane states that planar graphs are exactly graphs with basis number at most 2, and more generally, graphs embedded on a fixed surface are known to have bounded basis number. Generalising this, we prove that graphs excluding a fixed minor $H$ have bounded basis number. Our proof uses the Graph Minor Structure Theorem, which requires us to understand how basis number behaves in tree-decompositions. In particular, we prove that graphs of treewidth $k$ have basis number bounded by some function of $k$. We handle tree-decompositions using the proof framework developed by Bojańczyk and Pilipczuk in their proof of Courcelle's conjecture. Combining our approach with independent results of Miraftab, Morin and Yuditsky (2025) on basis number and path-decompositions, one can moreover improve our upper bound to a polynomial one: there exists an absolute constant $c>0$ such that every $H$-minor free graph has basis number $O(|H|^c)$.

Figures (5)

• Figure 1: Glueing of two bi-interface graphs. On the left two bi-interval graphs $\mathbf{G}_1, \mathbf{G}_2$ of arity $5$. On the right their glueing. The numbers correspond to the preimages of the interface vertices by the interface maps.
• Figure 2: Left: In green, a tree-decomposition $(T,\beta)$, and in red, the bags corresponding to a subset $X\subseteq V(T)$. Right: In blue, the quotient $(T/X, \beta_X)$.
• Figure 3: Configuration in the proof of Lemma \ref{['lem: 58-general']}. The dotted rectangle represents $e_{\ell}$, the region in light blue represents the vertices of $K$, and the purple regions represent $W_{\ell}$. Note that the hyperedges $f_j$ are not necessarily disjoint in general.
• Figure 4: Configuration of Lemma \ref{['lem: connecting-path-surface']}.
• Figure 5: Top: In dark green, a graph $G_0$ embedded in the orientable surface of genus $2$. In orange a collection of pairwise disjoint closed disks which only intersect $G_0$ in $V(G_0)$. Note that $D_1,D_3$ and $D_4$ are contained in the same face of $G_0$. Bottom: The graph $\Gamma$ constructed with respect to the disks $D_1,D_2,D_3,D_4$ is represented in thick, magenta. The graph $H_\Gamma$ defined later in the proof corresponds to the union of the magenta graph and the orange cycles.

Theorems & Definitions (97)

• Theorem 1.1: Mac Lane's planarity criterion, maclane1937planar
• Theorem 1.2: LM24
• Theorem 1.3
• Corollary 1.4
• Theorem 1.5
• Theorem 1.6
• Theorem 1.7: miraftab2026pathwidth
• Corollary 1.8: miraftab2026pathwidth and our results
• Theorem 1.9: miraftab2026pathwidth
• Corollary 1.10