Basis Number of Graphs Excluding Minors

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 of bounded genus 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 $O(|H|^c)$.

💡 Research Summary

The paper investigates the “basis number” of a finite graph G, defined as the smallest integer k such that the cycle space of G can be generated by a family of cycles in which every edge participates in at most k cycles. This parameter generalises Mac Lane’s classic planarity criterion, which states that planar graphs are exactly those with basis number at most 2. Earlier work showed that graphs embeddable on a surface of genus g have basis number O(log g), and that cliques have basis number 3, while some graphs have arbitrarily large basis number.

The central question addressed is whether the bounded‑basis‑number property extends to any proper minor‑closed class: does every graph that excludes a fixed minor H have basis number bounded by a function depending only on |H|? The authors answer affirmatively. Their main theorem (Theorem 1.3) asserts that there exists a function f₁.₃ such that for every graph H and every H‑minor‑free graph G, we have bn(G) ≤ f₁.₃(|H|). The proof initially yields a double‑exponential bound f₁.₃(t)=2^{2·O(t²)}. However, by combining their framework with independent recent work of Miraftab, Morin, and Yuditsky (2025), the bound can be improved to a polynomial one: there is an absolute constant c (≤ 32210) such that bn(G)=O(|H|^{c}) for every H‑minor‑free graph G.

The proof strategy relies heavily on the Robertson–Seymour Graph Minor Structure Theorem. According to this theorem, any H‑minor‑free graph can be assembled from four building blocks:

1. A graph embedded on a fixed surface (bounded genus);
2. A bounded number of vortices of bounded width glued inside faces of the embedding;
3. A bounded number of apices (vertices with arbitrary connections);
4. Clique‑sums of bounded order that combine the above pieces.

The authors analyze each operation’s effect on the basis number. For surface‑embedded graphs, they invoke the known O(log g) bound (Theorem 1.2). Adding ℓ apices increases the basis number by at most 2ℓ (Lemma 2.4). Handling vortices requires replacing “virtual” edges added in the torso of a tree‑decomposition by actual paths in the original graph; the congestion of the resulting path system must be controlled. The most delicate part is the behavior under bounded‑order clique‑sums, which they treat via tree‑decompositions of bounded adhesion.

Two technical results about tree‑decompositions are proved:

Theorem 1.5 – Every graph of treewidth k has basis number at most f₁.₅(k) = 2^{2·O(k²)}.
Theorem 1.6 – Let 𝔊 be a monotone class of graphs each with basis number ≤ b. If a graph G admits a tree‑decomposition of adhesion ≤ k whose torsos all belong to 𝔊, then bn(G) ≤ f₁.₆(b,k) = b·2^{2·O(k²)}.

Both proofs adapt the framework developed by Bojańczyk and Pilipczuk for Courcelle’s conjecture on MSO‑definable tree‑decompositions. The key idea is to select, for each bag, a low‑congestion cycle basis of its torso, then replace each virtual edge of the torso by a concrete path in the original graph. The collection of all such replacement paths must have bounded congestion. To guarantee this, the authors first handle path‑decompositions (bounded‑adhesion, linear layout) using Simon’s Factorisation Forest Theorem, which yields regularity conditions on how adhesions interact. After deleting a bounded number of vertices, these conditions produce the required low‑congestion path system, enabling the application of a general “tree‑decomposition preservation” theorem (Theorem 3.1).

The double‑exponential bounds arise primarily from the path‑width stage, where Simon’s theorem introduces a tower‑type growth. Recent independent work (Miraftab, Morin, Yuditsky) shows that graphs of pathwidth k actually have basis number at most 4k³, a linear‑in‑k bound. Incorporating this result collapses the double‑exponential to a polynomial. Consequently, they obtain:

• Corollary 1.8: Any graph of treewidth k has basis number O(k⁵).
• Theorem 1.9: If a graph admits a path‑decomposition of adhesion k whose parts each have basis number ≤ b, then bn(G) ≤ b + O(k log² k).

Combining Theorem 1.9 with the refined version of Theorem 1.6 yields f₁.₆(b,k) = O((b + k log² k)·k⁴). Using recent polynomial‑size bounds for the Graph Minor Structure Theorem (Gorczy, Seweryn, Wiederrecht), the authors finally derive the polynomial bound for H‑minor‑free graphs (Corollary 1.10).

The paper concludes with examples of graph families with unbounded basis number, discusses the tightness of their results, and outlines open problems such as determining the exact dependence on |H|, extending the notion to directed graphs, and exploring algorithmic aspects (e.g., constructing low‑congestion bases efficiently).

In summary, the work establishes that the basis number—a natural measure of how “economically” the cycle space can be generated—is uniformly bounded across any proper minor‑closed class. It showcases a powerful synthesis of deep structural graph theory (the Graph Minor Structure Theorem), algebraic graph concepts (cycle spaces), and logical methods (MSO‑definable decompositions), and it opens the door to further investigations of basis‑related parameters in both theoretical and applied settings.