REVIEW 1 major objections 2 minor 1 cited by
Separation profiles of free products
T0 review · 1 major / 2 minor · reviewed 2026-05-08 · grok-4.3
Pith's one-line read The separation and treewidth profiles of graphs are asymptotically equivalent.
desk verdict Hume deduces equivalence of separation and treewidth profiles from Dvorak-Norin to resolve an open question and computes them for tree-graded graphs including free products, but the infinite-graph application needs a clear justification. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The asymptotic equivalence between the separation profile and the treewidth profile, which carries the argument by allowing transfer of results and computations to composite graph structures.
What would settle it
A graph where the separation profile and treewidth profile have asymptotically different growth rates, such as a specific Cayley graph of a free product, would show the equivalence fails.
Extended reading notes
Core claim
We deduce that the separation and treewidth profiles of graphs are asymptotically equivalent. As an application, we calculate the separation profiles of Cayley graphs of tree-graded graphs in terms of their pieces. Examples include Cayley graphs of free products of finitely generated groups.
Load-bearing premise
The known theorem about graph separators applies directly to infinite Cayley graphs of tree-graded graphs.
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The manuscript deduces from a theorem of Dvorak--Norin that the separation and treewidth profiles of graphs are asymptotically equivalent, resolving a question of Huang--Hume--Kelly--Lam. As an application, it calculates the separation profiles of Cayley graphs of tree-graded graphs in terms of their pieces, with examples including Cayley graphs of free products of finitely generated groups.
Significance. If the deduction is justified, the paper resolves an open question on asymptotic equivalence of these profiles and supplies explicit formulas for separation profiles in the class of tree-graded graphs. The approach of reducing the core claim to an existing theorem is efficient; the value lies in the application to free products and the verification that the equivalence carries over to the infinite setting.
major comments (1)
- [Main deduction / application to infinite graphs] The central deduction from the Dvorak--Norin theorem: the manuscript states that the theorem directly yields asymptotic equivalence for the graphs considered, including infinite Cayley graphs of tree-graded graphs. However, treewidth and separation results are typically formulated for finite graphs; an explicit lifting argument (e.g., showing that sequences of finite subgraphs or balls witnessing one profile can be chosen to witness the other, or that the suprema/limits commute with the equivalence) is required to pass the relation to the infinite case. Without such a lemma, the claim for Cayley graphs rests on an unverified approximation property.
minor comments (2)
- [Abstract] The abstract and introduction could more explicitly state the form of the separation profile (e.g., the precise asymptotic expression in terms of the pieces) rather than only describing it as 'in terms of their pieces.'
- [Introduction] Notation for profiles (separation profile vs. treewidth profile) should be introduced with a brief reminder of the definitions from Huang--Hume--Kelly--Lam to make the deduction self-contained for readers.
Simulated Author's Rebuttal
We thank the referee for their careful reading of the manuscript and for identifying the need for an explicit argument when extending the Dvorak--Norin theorem to infinite graphs. We address the major comment below.
read point-by-point responses
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Referee: The central deduction from the Dvorak--Norin theorem: the manuscript states that the theorem directly yields asymptotic equivalence for the graphs considered, including infinite Cayley graphs of tree-graded graphs. However, treewidth and separation results are typically formulated for finite graphs; an explicit lifting argument (e.g., showing that sequences of finite subgraphs or balls witnessing one profile can be chosen to witness the other, or that the suprema/limits commute with the equivalence) is required to pass the relation to the infinite case. Without such a lemma, the claim for Cayley graphs rests on an unverified approximation property.
Authors: We agree that an explicit lifting argument improves clarity, even though the profiles for infinite graphs are defined via limits of the corresponding quantities on finite balls or subgraphs. Because the Dvorak--Norin theorem applies to every finite graph and the finite balls in a Cayley graph of a tree-graded graph inherit the tree-graded structure (with the same piece-wise bounds), the asymptotic equivalence passes to the infinite case. To make this rigorous, we will add a short lemma in Section 2 that shows the suprema and limits commute with the equivalence relation when the underlying finite graphs satisfy the hypotheses of Dvorak--Norin. This addresses the referee's concern without altering the main claims. revision: yes
Circularity Check
No circularity; central claim deduced from external Dvorak--Norin theorem
full rationale
The paper states it deduces asymptotic equivalence of separation and treewidth profiles directly from the theorem of Dvorak--Norin (an external result by unrelated authors). The application to Cayley graphs of tree-graded graphs and free products is presented as a consequence of this deduction. No self-definitional steps, fitted inputs renamed as predictions, load-bearing self-citations, uniqueness theorems imported from the authors' prior work, or ansatzes smuggled via citation appear in the abstract or described derivation. The resolved question of Huang--Hume--Kelly--Lam is merely the target application, not a premise that reduces the argument to self-reference. The derivation chain is therefore self-contained against the cited external benchmark.
Assumptions & free parameters
assumptions (1)
- domain assumption Theorem of Dvorak--Norin on separation and treewidth profiles
Cite this review
Pith. "Pith review of Separation profiles of free products." pith.science (2026). https://pith.science/paper/2604.24462
@misc{pith2026260424462,
author = {Pith},
title = {Pith review of: Separation profiles of free products},
year = {2026},
howpublished = {\url{https://pith.science/paper/2604.24462}},
note = {Machine review of arXiv:2604.24462}
}
read the original abstract
We deduce from a theorem of Dvorak--Norin that the separation and treewidth profiles of graphs are asymptotically equivalent, resolving a question of Huang--Hume--Kelly--Lam. As an application, we calculate the separation profiles of Cayley graphs of tree-graded graphs in terms of their pieces. Examples of tree-graded graphs include Cayley graphs of free products of finitely generated groups.
Forward citations
Cited by 1 Pith paper
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Separation profiles of hyperbolic planar and apex-minor-free graphs
The separation profile of any δ-hyperbolic planar or apex-minor-free graph grows at most as C log n, answering Benjamini–Schramm–Timár affirmatively.
Reference graph
Works this paper leans on
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[1]
Hans L. Bodlaender. Treewidth: Characterizations, applications, and computations. In Fedor V. Fomin, editor, Graph-Theoretic Concepts in Computer Science , pages 1--14, Berlin, Heidelberg, 2006. Springer Berlin Heidelberg
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[2]
On the separation profile of infinite graphs
Itai Benjamini, Oded Schramm, and \'A dam Tim \'a r. On the separation profile of infinite graphs. Groups Geom. Dyn. , 6(4):639--658, 2012
work page 2012
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[3]
Poincar\'e profiles of lamplighter diagonal products
Corentin Le Coz. Poincar\'e profiles of lamplighter diagonal products. Preprint, arXiv:2007.04709 , 2020
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[4]
Tree-graded spaces and asymptotic cones of groups
Cornelia Dru t u and Mark Sapir. Tree-graded spaces and asymptotic cones of groups. Topology , 44(5):959--1058, 2005. With an appendix by Denis Osin and Mark Sapir
work page 2005
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[5]
Treewidth of graphs with balanced separations
Zdenek Dvorak and Sergey Norin. Treewidth of graphs with balanced separations. J. Comb. Th. , Series B. 137, 2014
work page 2014
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[6]
Separation profile, isoperimetry, growth and compression
Antoine Gournay and Corentin Le Coz. Separation profile, isoperimetry, growth and compression. Annales de l'Institut Fourier , 73(4):1627--1675, 2023
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[7]
Separation profiles of graphs of fractals
Valeriia Gladkova and Verna Shum. Separation profiles of graphs of fractals. Journal of Topology and Analysis , 13(04):1111--1123, 2021
work page 2021
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[8]
Wanying Huang, David Hume, Samuel J. Kelly, and Ryan Lam. A coarse geometric approach to graph layout problems. To appear in the Journal of Graph Theory . Preprint, arXiv:2312.07015 , 2023
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[9]
David Hume and John M. Mackay. Connecting conformal dimension and Poincaré profiles. Preprint, arXiv:2511.10469 , 2025
2025
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[10]
Mackay, and Romain Tessera
David Hume, John M. Mackay, and Romain Tessera. Poincar\' e profiles of groups and spaces. Rev. Mat. Iberoam. , 36(6):1835--1886, 2020
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[11]
Mackay, and Romain Tessera
David Hume, John M. Mackay, and Romain Tessera. Poincar\'e profiles of L ie groups and a coarse geometric dichotomy. Geom. Funct. Anal. , 32:1063--1133, 2022
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Quasi-isometries between groups with infinitely many ends
Panos Papasoglu and Kevin Whyte. Quasi-isometries between groups with infinitely many ends. Comment. Math. Helv. , 77(1):133--144, 2002
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[13]
Neil Robertson and P. D. Seymour. Graph minors. II. Algorithmic aspects of tree-width. Journal of Algorithms , 7(3):309--332, 1986
1986
Reviewed May 8, 2026 · model on record in the stance chip above.
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