Subpolynomial Treewidth in Graph Theory

• Subpolynomial treewidth is defined for graph classes where the treewidth grows slower than any polynomial in the number of vertices, distinguishing near-tree structures from highly interconnected ones.
• Methods like star-forest decompositions, slim pair/barrier arguments, and balanced separators are key to achieving and analyzing these subpolynomial bounds.
• This structural property leads to significant algorithmic benefits, including efficient MSO property testing and quasi-polynomial solutions for hard problems in restricted graph classes.

Subpolynomial treewidth refers to classes of graphs whose treewidth grows with the number of vertices $n$ at a rate strictly lower than any polynomial in $n$. That is, for such classes, $\operatorname{tw}(G) = n^{o(1)}$ for every $n$-vertex graph $G$ in the class. Treewidth—an essential parameter in both structural graph theory and algorithmic graph theory—typically distinguishes “tree-like” connectivity from highly entangled, grid-like graphs. Subpolynomial treewidth marks a threshold between classes that remain computationally tractable for hard problems via dynamic programming on tree decompositions, and those that do not.

1. Definitions and Preliminaries

Treewidth $\operatorname{tw}(G)$ of a graph $G$ measures, informally, how close $G$ is to being a tree; formally, it is the minimum width among all tree decompositions. Bounded treewidth ($\operatorname{tw}(G)\le c$ for fixed $c$) is a strong structural constraint, and classes with bounded treewidth admit polynomial-time algorithms for many central NP-complete problems.

A class $\mathcal{G}$ is said to have subpolynomial treewidth if $\operatorname{tw}(G) \le n^{\varepsilon}$ for every $G\in\mathcal{G}$ and every $\varepsilon>0$, given $n$ (the number of vertices) is sufficiently large (Chudnovsky et al., 21 Dec 2025). Typical regimes of interest include logarithmic, polylogarithmic, and functions of the form $2^{O(\log^{1-\varepsilon} n)}$.

Related Obstructions and Forbidden-Pattern Classes

Classes with subpolynomial treewidth are often defined by excluding certain induced subgraphs, minors, or induced minors; for example, forbidding grid-like structures (e.g., $t\times t$ hexagonal grids), large complete bipartite subgraphs ($K_{t,t}$), or specific path configurations such as “thetas” or “three-path-configurations” (Abrishami et al., 2021, Chudnovsky et al., 21 Dec 2025).

2. Main Theorems: Structural Barriers and Tight Bounds

The foundational result for subpolynomial treewidth in recent research is the theorem of Chudnovsky, Codsi, Fischer, and Lokshtanov:

For every $t\in\mathbb{N}$, there exist $\varepsilon\in(0,1]$ and $c\in\mathbb{N}$ such that every $n$-vertex $\{K_{t,t},\,W_{t\times t}\}$-induced-minor-free graph with clique number $<t$ satisfies $$\operatorname{tw}(G)\le 2^{c (\log n)^{1-\varepsilon}}$$ (Chudnovsky et al., 21 Dec 2025).

Consequently, such classes achieve subpolynomial treewidth, and for every fixed $\varepsilon>0$, $\operatorname{tw}(G)\le n^{\varepsilon}$ for large enough $n$.

For more restricted hereditary classes, even stronger results are obtained:

• For (theta, triangle)-free graphs, $\operatorname{tw}(G)\le c\log |V(G)|$ for an absolute constant $c$, and this is optimal (Abrishami et al., 2021).
• Sparse outerstring graphs with arboricity $\alpha$ attain $\operatorname{tw}(G)=\Theta(\alpha\log n)$; $t$-biclique-free outerstring graphs achieve $\operatorname{tw}(G)=O(t\log t\log n)$ (An et al., 25 Jun 2024).
• When the maximum degree is $O(\log n)$ and planar induced minors and $K_{t,t}$ subgraphs are excluded, treewidth is polylogarithmic in $n$ (Bonnet et al., 2023).

The following table synthesizes principal regimes:

Forbidden Structures/Class	Treewidth Bound	Cited Result
$(\theta, K_3)$-free	$O(\log n)$	(Abrishami et al., 2021)
$\{K_{t,t}$, $W_{t\times t}\}$-induced-minor-free, $\omega(G)<t$	$2^{O(\log^{1-\varepsilon} n)}$	(Chudnovsky et al., 21 Dec 2025)
Outerstring, $t$-biclique-free	$O(t\log t \log n)$	(An et al., 25 Jun 2024)
Planar induced minor-free, $K_{t,t}$-free, $\Delta=O(\log n)$	$O((\log n)^{f(k,t)})$	(Bonnet et al., 2023)

3. Methodologies and Proof Techniques

Recent advances leverage an overview of separator theory, sparse graph decompositions, star-forest coverings, and robust “slim pair/barrier” arguments.

A. Star-forest Decomposition:

Graph edge-partitioning into star forests (edge-disjoint collections where each component is a star) is instrumental—especially for $K_{t,t}$-induced-minor-free graphs. Combined with bounded clique number, such partitionings allow a reduction to “dimension” one-parameter arguments via contraction and induction (Chudnovsky et al., 21 Dec 2025, Bonnet et al., 2023).

B. Slim Pair/Barrier Arguments:

A pair $(a, b)$ is “$q$-slim” if there are not $q$ disjoint, mutually anticomplete $a$–$b$ paths. The construction of “barriers”—small separator subgraphs that must be traversed by all such paths—helps establish the existence of small separators recursively, giving control over treewidth growth (Chudnovsky et al., 21 Dec 2025).

C. Balanced Separators:

Classic connections (Robertson–Seymour, Harvey–Wood) link the existence of small balanced separators to treewidth bounds. If every weight function admits a separator of subpolynomial size, the global treewidth is likewise subpolynomial.

D. High-Degree Sparsifiers and Expander Embeddings:

Degree-reduction and well-linked set arguments, sometimes realized via the ARV cut-approximation or the Chekuri–Chuzhoy sparsifier, are used to round out the analysis—reducing complex connectivity to tractable pieces of small degree or expansion (Chekuri et al., 2013, Bonnet et al., 2023).

4. Canonical Extremal Examples and Optimality

Tightness of logarithmic or subpolynomial treewidth is validated by explicit constructions:

• Sintiari–Trotignon’s “layered wheels” yield (theta, triangle)-free graphs with $|V(G_\ell)|=2^{O(\ell)}$ yet $\operatorname{tw}(G_\ell)\ge \ell$, so $\operatorname{tw}(G_\ell)=\Theta(\log |V(G_\ell)|)$, demonstrating the asymptotic optimality of $O(\log n)$ bounds (Abrishami et al., 2021).
• For outerstring graphs, matching lower bounds are obtained by constructing families with $\alpha$-controlled arboricity achieving $\operatorname{tw}(G)=\Omega(\alpha \log(n/\alpha))$ (An et al., 25 Jun 2024).

These constructions underscore that the logarithmic barrier for treewidth is tight for many hereditary or geometric intersection classes with sufficient forbidden structure.

5. Algorithmic Consequences

Subpolynomial treewidth is both a structural and algorithmic accelerator. Central consequences include:

• For any class with treewidth $f(n) = n^{o(1)}$, any MSO-definable property is decidable in time $2^{O(f(n))}n^{O(1)}$, which is polynomial if $f(n) = O(\log n)$ (by Courcelle’s theorem and dynamic programming) (Abrishami et al., 2021).
• In $t$-biclique-free outerstring graphs, polynomial-time algorithms become available for Independent Set, Vertex Cover, Dominating Set, Feedback Vertex Set, and Coloring, leveraging treewidth at most $O(t\log t\log n)$. Exact, parameterized, and approximation algorithms with subexponential complexity become available for broader intersection classes (An et al., 25 Jun 2024).
• For planar induced minor and $K_{t,t}$-free classes with $\Delta(G)=O(\log n)$, such as those described in (Bonnet et al., 2023), polylogarithmic treewidth permits polynomial-time and quasi-polynomial-time algorithms for a host of hard problems.
• Treewidth-decomposition theorems with parameters $(h, r)$—where $h$ is the number of parts and $r$ the treewidth per part—yield improved bounds for Erdős-Pósa-type results and more efficient FPT algorithms for bidimensional parameters (Chekuri et al., 2013).

6. Limitations, Open Problems, and Future Directions

Several boundaries remain in subpolynomial treewidth theory:

• Role of Clique-Number Hypothesis: For the class of $\{K_{t,t}, W_{t\times t}\}$-induced-minor-free graphs, the lack of a uniform subpolynomial treewidth bound without a bounded clique-number assumption is an open issue. The clique-size constraint is needed to avoid blow-up constructions of high treewidth (Chudnovsky et al., 21 Dec 2025).
• Biclique-Free Constraints: The necessity of forbidding $K_{t,t}$ (or analogous dense bipartite structures) in obtaining treewidth bounds is critical. It is unknown whether similar results hold under strictly weaker conditions (Bonnet et al., 2023).
• Towards Linear or Polylogarithmic Treewidth: A conjecture posits that planar induced minor exclusion (alone) could imply $\operatorname{tw}(G)\le O(\Delta(G))$ for graphs with bounded degree; such a result would dramatically expand the classes where subpolynomial treewidth arises and potentially sharpen current algorithmic applications (Bonnet et al., 2023).
• Separation Complexity and Algorithmic Width: Whether the current log or polylog exponents can be reduced further, e.g., to $O(\log n)$ or even $O(\log n \cdot \operatorname{polylog}\log n)$ for certain natural classes, is a major open direction.
• Extension to Other Width Measures: Extending subpolynomial bounds from treewidth to related parameters such as pathwidth or carving-width, including in the directed graph or multi-dimensional paradigms, remains unexplored terrain (Chekuri et al., 2013).

7. Connections to Broader Graph-Structural Theory

Subpolynomial treewidth forms a boundary between “well-structured” sparse graph classes—amenable to dynamic programming and width-based algorithms—and classes exhibiting polynomial or linear treewidth growth, which preclude robust tractability for many classic problems.

Recent research synthesizes forbidden minor theory, separator characterizations, sparse decompositions, and the interplay of geometric and combinatorial obstructions. Foundational developments such as the Grid-Minor Theorem are bypassed in favor of local well-linkedness and expander-based modules, leading to tighter, more algorithmic-friendly width bounds in nontrivial graph classes (Chekuri et al., 2013, Chudnovsky et al., 21 Dec 2025). The hierarchy from bounded, logarithmic, polylogarithmic, to truly subpolynomial treewidth is now central in both the theoretical landscape and in practical algorithm design where structural graph constraints are imposed.