Quasi-isometries, contractions, and intersection graphs

Agelos Georgakopoulos; Chiara Molinari

REVIEW 1 major objections 5 minor 40 references

Quasi-isometries, contractions, and intersection graphs

T0 review · 1 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read Every graph that is quasi-isometric to a planar graph is obtained from planar graphs by finitely many bounded subdivisions and cover-intersection operations, and conversely.

desk verdict Strong, significant paper with a load-bearing lemma that is only sketched; the characterization is plausible and worth referee time. read the letter →

arxiv 2608.10164 v1 pith:ZDXG2WNM submitted 2026-08-10 math.CO math.GRmath.MG

Agelos Georgakopoulos , Chiara Molinari This is my paper · ORCID

classification math.COmath.GRmath.MG MSC 05C1005C6205C1251F3005C6305C76

keywords quasi-isometryquasi-planarintersectiongraphcover-intersectionstringcontractionminoredge-slidingbi-Lipschitzequivalence

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

Quasi-planar graphs are graphs that look planar from far away: each is quasi-isometric to a planar graph, so distances are preserved up to multiplicative and additive error. The paper's main theorem gives a purely combinatorial description of this coarse-geometric class: a graph is quasi-planar exactly when it can be obtained from a planar graph by repeating two local operations, subdividing every edge into a bounded-length path and forming the intersection graph of a family of connected subgraphs that cover the graph. The backward direction uses the known theorem that every string graph is quasi-isometric to a planar graph with universal constants, while the forward direction is new and runs through a graph operation the paper calls edge-sliding. If correct, the result means quasi-planarity is no longer only a metric notion: it is the closure of planar graphs under an explicit finite recipe.

What carries the argument

The two load-bearing constructions are the subdivision-cover-intersection operation and edge-sliding. The subdivision-cover-intersection operation takes a graph, subdivides each edge into a path of length at most 3, and then forms the cover-intersection graph: the vertices are a family of connected subgraphs that cover the base graph, and two vertices are adjacent when the corresponding subgraphs intersect. Edge-sliding is an equivalence relation on graphs sharing a vertex set: an edge can be moved by sliding one endpoint along another edge of the shared frame, and two graphs are edge-sliding twins when every difference can be removed this way; twins are 2-bi-Lipschitz equivalent, and every bi-Lipschitz equivalence between graphs on the same vertex set can be decomposed into finitely many edge-slidings. The proof's core lemma realizes a single edge-sliding by two subdivision-cover-intersection operations with cover sets of diameter at most 4, while bounded-diameter cover sets make the operation quasi-isometry-preserving (Corollary 3.3).

What would settle it

A countable graph that is quasi-isometric to a planar graph but is not an iterated subdivision-cover-intersection graph of any planar graph would refute Theorem 1.7. The square grid with both diagonals added to every cell is a concrete test case: it is non-planar, it is quasi-isometric to the planar grid, and the constructive proof must produce a finite SCIG derivation; if the number of iterations or the diameters of the cover sets grow without bound for larger and larger finite grids, the uniform finite-graph version fails.

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Extended reading notes

Core claim

The central claim is Theorem 1.7: a graph is quasi-isometric to a planar graph if and only if it is an iterated subdivision-cover-intersection-graph of a planar graph. Iterating means starting with a planar graph and, a finite number of times, first subdividing every edge into a path of length at most 3 and then taking the intersection graph of a family of connected subgraphs whose union is the whole graph. The paper also proves the more general Theorem 1.8, in which two graphs are quasi-isometric exactly when each is obtained from the other by such operations with cover sets of bounded diameter. The backward direction of Theorem 1.7 builds on the cited theorem that every string graph is quasi-isometric to a planar graph with universal constants; the forward direction is proved by introducing edge-sliding and showing that every bi-Lipschitz equivalence decomposes into edge-slidings, each of which is realized by two subdivision-cover-intersection steps. Along the way the paper shows that contraction minors of quasi-planar graphs are quasi-planar, and that tree-decompositions with bounded-diameter adhesions and quasi-planar induced bags preserve quasi-planarity.

Load-bearing premise

The load-bearing premise is the cited theorem, not reproved here, that every countable string graph is quasi-isometric to a planar graph with universal constants; the backward half of the main equivalence collapses if that theorem fails. The paper also assumes throughout that all graphs are countable.

Editorial extensions

If this is right

Quasi-planarity now has a finite combinatorial certificate: a graph is quasi-planar exactly when a finite sequence of bounded subdivisions and cover-intersection steps leads from a planar graph to it.
Every contraction minor of a quasi-planar graph is quasi-planar, with constants depending explicitly on the original quasi-isometry constants.
Tree-decompositions with bounded-diameter adhesions and quasi-planar induced bags produce quasi-planar graphs, so quasi-planarity composes along tree-like splittings.
A finitely generated group that splits over a finite subgroup into virtually-planar factors is itself virtually-planar.
Any graph quasi-isometric to a planar graph is bi-Lipschitz equivalent to a planar graph, so the metric and bi-Lipschitz notions of quasi-planarity coincide at the level of existence.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

Inference: If the string-graph theorem is extended from planar graphs to every minor-closed family, as the paper reports is plausible, the same SCIG characterization would give a uniform combinatorial description of every quasi-minor-closed class.
Inference: The characterization turns quasi-planarity into an existence problem with finite witnesses; for fixed constants, verifying a candidate SCIG derivation is a local combinatorial check, which may make quasi-planarity algorithmically recognizable in ways that the metric definition does not.
Inference: The paper leaves open whether subdivisions can be dropped from the characterization (Problem 7.1); a positive answer would reduce the description to iterated string graphs, making the generative recipe purely intersection-theoretic.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the letter

Paul — this is a serious paper and the main result is genuinely new: quasi-planar graphs are exactly the iterated subdivision-cover-intersection-graphs of planar graphs (Thm 1.7). It also proves contraction-minor closure (Thm 1.2), which was conjectured, and a tree-decomposition preservation theorem (Thm 1.10). The machinery — Voronoi cells, edge-sliding, the local-to-global glueing in Thm 4.3 — is well-chosen, and the authors are careful with explicit constants and constructive proofs. The edge-sliding notion in Section 6.1 is a nice independent tool; Lemma 6.9 (bi-Lipschitz iff iterated edge-sliding twins) cleanly packages the metric/combinatorial interface.

The soft spots are real but not fatal. Lemma 6.1, which reduces quasi-isometric graphs to bi-Lipschitz equivalence via one SCIG operation, is the hinge of the forward direction and its proof is a sketch: 'the required cover is natural' is doing a lot of work. A referee should ask for a full construction, especially for how merged star sets produce exactly the desired intersection graph without spurious edges, and how the bounded-diameter condition is maintained when contracted components are not stars. I suspect it is fillable, but as written it is a gap in the proof of the main theorem. Similarly, the countable assumption is stated in Section 2 but omitted from the abstract and Theorem 1.7, so the 'infinite graphs' claim overreaches. There's also a small constant typo in Theorem 4.3 (2AM^2r should presumably be 2M^2r + 6AM).

The dependence on Davies' theorem for the backward direction is clearly acknowledged and is not a flaw; the reverse implication is new. No circularity or data-fitting issues.

If I were the editor, I would send this to a serious referee. The main theorem is important enough, and the rest of the paper is detailed enough, that the missing Lemma 6.1 details are worth one revise-and-resubmit cycle rather than a desk reject. The paper is for researchers in coarse graph theory, geometric group theory, and graph minors. I'd bring it to a reading group.

Referee Report

1 major / 5 minor

Summary. The paper develops tools in coarse graph theory around cover-intersection graphs and a new 'edge-sliding' equivalence relation, and uses them to prove a combinatorial characterization of quasi-planar graphs. Theorem 1.7 states that a graph is quasi-isometric to a planar graph if and only if it is an iterated subdivision-cover-intersection-graph (SCIG) of a planar graph. The backward direction builds on Davies' theorem that countable string graphs are quasi-planar; the forward direction is derived from the more general Theorem 1.8, which asserts that two graphs are quasi-isometric if and only if each can be obtained from the other by a bounded number of bounded-diameter SCIG operations. The paper also proves that quasi-planarity is preserved under contraction minors (Theorem 1.2), gives a tree-decomposition gluing theorem (Theorem 1.10/4.3), and derives applications to finitely generated groups, including a quasi-isometry statement for finite-index subgroups. The main technical work is in Section 6, where Lemma 6.1 reduces quasi-isometry to bi-Lipschitz equivalence, Lemma 6.9 connects bi-Lipschitz equivalence to iterated edge-sliding twins, and Lemma 6.10 performs a four-step construction realizing each edge-sliding step by two SCIG operations.

Significance. If the proofs are completed as intended, this is a substantial contribution: it gives a purely combinatorial, generative description of a whole quasi-isometry class, with explicit quantitative versions for families of finite graphs. The paper is careful with constants and constructive bounds, and the edge-sliding notion is likely to be of independent interest. The contraction-minor closure, the tree-decomposition theorem, and the group-theoretic corollaries are concrete, falsifiable consequences that go beyond the main characterization. However, the forward direction of the central theorem currently rests on a lemma whose proof is only sketched, so the main characterization is conditional on additional work.

major comments (1)

[§6, Lemma 6.1] The proof of Lemma 6.1 is only a sketch, and it is load-bearing: the forward direction of Theorem 1.8, and hence the forward direction of Theorem 1.7, begins with this reduction. The text says the required cover is 'natural' and that edge contractions and clique attachments 'can be realized' by merging and adding cover sets in a subdivision of G, but it does not specify which edges of G are subdivided, how the merged cover sets are defined when a contracted component from Proposition 5.2 is not a star, why the resulting intersection graph has exactly the edge set of the target graph (no missing or spurious edges), and why every cover set remains connected with diameter bounded by a function of M and A only. Without these details the claimed reduction is not checkable. I recommend proving this lemma in full, or replacing the forward direction with a direct construction.

minor comments (5)

[§4, proof of Theorem 4.3] In the lower-bound estimate for d_G(x,y), the printed constant '2AM^2 r' should be '2M^2 r': the preceding inequality gives d_G(˚x_i,˚x_{i+1}) ≤ 2M^2 r + 6AM, without an extra factor of A. As written, the bound fails when A = 0.
[§2 and abstract] Section 2 states that all graphs are assumed countable, and the abstract says the results apply to infinite graphs and to finite families with uniform constants, but the statements of Theorem 1.7 and Corollary 1.6 omit the word 'countable' for infinite graphs. This qualifier should appear in the theorem statements.
[§6.2, proof of Theorem 1.8] The phrase 'by Lemma 6.1, we have reduced to the case where G and H are bi-Lipschitz equivalent' should be made explicit, since Theorem 1.8(ii) requires both memberships G ∈ SCIG_n(H) and H ∈ SCIG_n(G); the reduction uses Lemma 6.1 once in each direction. As written, the proof only describes one application.
[§6.3, proof of Lemma 6.10, Step 2] The assignment of green labels to edges of E(G2)\E(A) is said to 'involve a choice' for edges of the second type. Please add a sentence explaining explicitly why the truth of (13) and the final isomorphism G4 ≅ H are independent of these choices; the current text asserts this but does not justify it.
[§6.1, Lemma 6.9] The proof of (i)⇒(ii) in Lemma 6.9 begins 'suppose G and H are bi-Lipschitz equivalent via the identity map'. For a general bi-Lipschitz equivalence one should first identify the vertex sets by the given bijection; this is standard but should be stated.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the forward direction rests on new SCIG constructions and the backward direction on Davies' external theorem.

full rationale

The paper's central claim (Theorem 1.7) is not circular. The backward direction is explicitly conditional on Davies' theorem (Theorem 1.1), an external result cited from [14,16]; the induction in Corollary 1.6 transfers quasi-planarity along cover-intersection-graphs using Corollary 1.5, whose proof is a self-contained construction (Theorems 1.3 and 1.4). The forward direction is a new chain: Lemma 6.1 reduces quasi-isometry to bi-Lipschitz equivalence via an elementary shallow-contraction/clique-attachment observation (Proposition 5.2, proved in the paper, and Proposition 5.1, cited to [25]); Lemmas 6.9 and 6.10 then give a constructive two-step SCIG realization of edge-sliding twins. None of these steps defines SCIG in terms of quasi-isometry or fits a parameter to the claimed output. The one author-self-citation (Proposition 5.1 from [25]) is a parameter-free, elementary observation and thus counts as independent support rather than circularity. The proof of Lemma 6.1 is only sketched, which is a proof-completeness concern, not a circularity: the reduction is not by construction equal to the theorem's conclusion. No fitted-input-called-prediction, uniqueness-imported-from-authors, or ansatz-smuggled-in-via-citation pattern occurs.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

No data fitting or fitted constants appear; the paper's constants are chosen for the construction, not fitted to examples. The edge-sliding operation is a mathematical definition rather than an empirical entity, so no invented physical entities are introduced. Load-bearing external inputs are Davies' theorem, countability, standard tree-decomposition facts, and MacManus' group theorem.

assumptions (4)

domain assumption Every graph in this paper is assumed countable (Section 2).
Main theorems state 'a graph' without this qualifier; the abstract says 'infinite graphs'. Constructions such as Voronoi partitions and cover-intersection graphs may require countability.
domain assumption Theorem 1.1: every countable string graph is (M,A)-quasi-isometric to a planar graph for universal constants (Davies [16], Chang, Conroy, Tan, Zheng [14]).
Used at page 2 and in proofs of Corollary 1.6, Theorems 1.2 and 3.2. It is an external deep result, not reproved here.
standard math Standard tree-decomposition facts, including separation properties (Diestel [20, Lemma 12.3.1]) and the radius-enlargement lemma (Albrechtsen et al. [2, Lemma 3.3]).
Invoked in Lemmas 4.1 and 4.2 to preserve tree-decompositions and bound distances.
domain assumption MacManus' theorem [36, Corollary D]: a finitely generated group is quasi-planar iff it is virtually-planar.
Used only in Corollary 1.11 to translate quasi-planarity of a Cayley graph into virtual planarity.

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Cite this review

Pith. "Pith review of Quasi-isometries, contractions, and intersection graphs." pith.science (2026). https://pith.science/paper/ZDXG2WNM

@misc{pith2026260810164,
  author       = {Pith},
  title        = {Pith review of: Quasi-isometries, contractions, and intersection graphs},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZDXG2WNM}},
  note         = {Machine review of arXiv:2608.10164}
}

abstract

We prove that a graph $G$ is quasi-planar - i.e. quasi-isometric to a planar graph - if and only if it can be obtained by iterating the following two operations a bounded number of times: a) subdividing each edge into a path of bounded length, and b) taking the intersection graph of a family of connected subgraphs covering $G$. This applies both to infinite graphs, and to families of finite graphs with uniform constants. The backward implication relies on, and generalises, a deep result of Davies, partly proved independently by Chang, Conroy, Tan & Zheng, saying that every string graph is quasi-planar. The forward implication requires new ideas. As a byproduct of our proofs, we deduce that every contraction minor of a quasi-planar graph is quasi-planar. Moreover, if $G$ admits a tree-decomposition with adhesions of bounded diameter and quasi-planar induced bags, then $G$ is itself quasi-planar. Our results apply to other graph classes as well, and we offer various tools for understanding quasi-isometries as well as bi-Lipschitz equivalences between graphs.

Figures

Figures reproduced from arXiv: 2608.10164 by the authors.

**Figure 2.** To construct H, we first attach green labels to some edges of G. We record the following rather obvious observation: for every x, y ∈ V , we have xy ∈ E(H) if and only if xy ∈ E(G ∩ H) or there is a green label (x, y) in G0. (10) Indeed, for the backward implication, note that green labels take values in T(EG)∪EH ⊆ E(H). For the forward implication, if xy ∈ E(G∩ H) we are done. Otherwise, xy ∈ EH and, by constructio… view at source ↗

**Figure 3.** G1 is a subdivision of G0 Step 2: Let C1 be the cover of M induced by the cover of G ∩ H given by edgebags and vertex-bags ( [PITH_FULL_IMAGE:figures/full_fig_p018_3.png] view at source ↗

Figures from the paper (4 more)

**Figure 4.** Figure 4: The subgraphs of G1 in C1 induce A as cover-intersection-graph. (left)). We remark that for this extension it does not matter whether we slide xy to zx or the other way round. This process results in a cover of G1 and we denote with G2 the resulting cover-intersection-…

**Figure 5.** Figure 5: G1 is fully covered and G2 is the resulting cover-intersection-graph. We now assign to each edge in E(G2) \ E(A) a single green label. While defining this assignment, we will verify that it satisfies the following two properties: if there is a green label (z, x) in G2,…

**Figure 6.** Figure 6: G3 is a subdivision of G2. Step 4: We now cover A with clique-bags {Kv}v∈V , where each Kv is the clique induced by the vertices of A bearing v as a red label ( [PITH_FULL_IMAGE:figures/full_fig_p020_6.png]

**Figure 7.** Figure 7: clique-bags are extended to cover G3 in full. Note that V (G4) ∼= V , via the map that sends every vertex v ∈ V to K¯ v . We claim that this map is in fact an isomorphism from H to G4: For every x, y ∈ V , we have K¯ xK¯ y ∈ E(G4) if and only if xy ∈ E(G ∩ H) or there …

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

40 extracted references · 23 canonical work pages

[1]

Abrishami, M

T. Abrishami, M. Briański, J. Davies, X. Du, J. Masaříková, P. Rzążewski, and B. Walczak. Burling graphs in graphs with large chromatic number. InProceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3978–3998, 2026

work page 2026
[2]

Albrechtsen, R

S. Albrechtsen, R. Diestel, A.-K. Elm, E. Fluck, R. W. Jacobs, P. Knappe, and P. Wollan. A structural duality for path-decompositions into parts of small radius. Innovations in Graph Theory, 3:207–246, 2026

2026
[3]

Albrechtsen, M

S. Albrechtsen, M. Distel, and A. Georgakopoulos. ExcludingK2,t as a fat minor. arXiv:2510.14644

arXiv
[4]

Albrechtsen, M

S. Albrechtsen, M. Distel, and A. Georgakopoulos. Small counterexamples to the fat minor conjecture. arXiv:2601.05761

arXiv
[5]

A coarse block-cutvertex tree-decomposition

S. Albrechtsen and A. Georgakopoulos. A coarse block-cutvertex tree- decomposition. arXiv:2607.07030

work page Pith review arXiv
[6]

Albrechtsen, R

S. Albrechtsen, R. Jacobs, P. Knappe, and P. Wollan. A characterisation of graphs quasi-isometric toK 4-minor-free graphs.Combinatorica, 45(61), 2025

work page 2025
[7]

Berger and P

E. Berger and P. Seymour. Bounded diameter tree-decompositions.Combinatorica, 44(1):659–674, 2024

work page 2024
[8]

Blazej, M

V. Blazej, M. Pilipczuk, and E. Protopapas. A coarse Menger’s Theorem for planar and bounded genus graphs. arXiv:2605.11112

arXiv

Show all 40 references

[9]

Bonamy, N

M. Bonamy, N. Bousquet, L. Esperet, C. Groenland, C.-H. Liu, F. Pirot, and A. Scott. Asymptotic Dimension of Minor-Closed Families and Assouad-Nagata Dimension of Surfaces.J. Eur. Math. Soc., 26(10):3739–3791, 2023

2023
[10]

Bonnet and R

É. Bonnet and R. Hickingbotham. Induced minors and region intersection graphs. Innovations in Graph Theory, 2:313–327, 2025

2025
[11]

Bonnet, H

É. Bonnet, H. Le, Ma. Pilipczuk, and Mi. Pilipczuk. Coarse Balanced Separators in Fat-Minor-Free Graphs. arXiv:2604.11318

arXiv
[12]

Brandstädt, F

A. Brandstädt, F. F. Dragan, H.-O. Le, and V. B. Le. Tree spanners on chordal graphs: complexity and algorithms.Theor. Comput. Sci., 310(1):329–354, 2004

2004
[13]

Catusse, V

N. Catusse, V. Chepoi, and Y. Vaxès. Planar hop spanners for unit disk graphs. In C. Scheideler, editor,Algorithms for Sensor Systems, pages 16–30. Springer Berlin Heidelberg, 2010

2010
[14]

Chang, J

H.-C. Chang, J. Conroy, Z. Tan, and D. W. Zheng. O(1)-Distortion Planar Emula- tors for String Graphs. arXiv:2510.21700

arXiv
[15]

Chepoi, F

V. Chepoi, F. F. Dragan, I. Newman, Y. Rabinovich, and Y. Vaxès. Constant Ap- proximation Algorithms for Embedding Graph Metrics into Trees and Outerplanar Graphs.Discrete & Computational Geometry, 47(1):187–214, 2012

2012
[16]

J. Davies. String graphs are quasi-isometric to planar graphs. arXiv:2510.19602
[17]

Davies, A

J. Davies, A. Georgakopoulos, M. Hatzel, and R. McCarty. Strongly Sublinear Separators and Bounded Asymptotic Dimension for Sphere Intersection Graphs. In O. Aichholzer and H. Wang, editors,41st International Symposium on Computa- tional Geometry (SoCG 2025), volume 332 ofLeib...

2025
[18]

Davies, R

J. Davies, R. Hickingbotham, F. Illingworth, and R. McCarty. Fat minors cannot be thinned (by quasi-isometries).Analysis and Geometry in Metric Spaces, 14(1), 2026

2026
[19]

Diestel, R

R. Diestel, R. W. Jacobs, P. Knappe, and J. Kurkofka. Canonical graph decompo- sitions via coverings. arXiv:2207.04855

arXiv
[20]

Springer-Verlag, 2025

Reinhard Diestel.Graph Theory(6th edition). Springer-Verlag, 2025. Electronic edition available at: http://www.math.uni-hamburg.de/home/diestel/books/graph.theory

2025
[21]

Distel, U

M. Distel, U. Giocanti, J. Hodor, C. Legrand-Duchesne, and P. Micek. A coarse Gallai theorem. arXiv:2601.18439
[22]

Esperet and U

L. Esperet and U. Giocanti. Coarse geometry of quasi-transitive graphs beyond planarity.Europ. J. Comb., 31(2):P2.41, 2024

2024
[23]

Fujiwara and P

K. Fujiwara and P. Papasoglu. A coarse-geometry characterization of cacti. arXiv:2305.08512

arXiv
[24]

Fujiwara and P

K. Fujiwara and P. Papasoglu. Asymptotic dimension of planes and planar graphs. Trans. Am. Math. Soc., 374:8887–8901, 2021

2021
[25]

Georgakopoulos and P

A. Georgakopoulos and P. Papasoglu. Graph minors and metric spaces.Combina- torica, 45:33, 2025

2025
[26]

Georgakopoulos and F

A. Georgakopoulos and F. Vigolo. Triangulating surfaces quasi-isometrically. arXiv:2603.21189

arXiv
[27]

Godsil and G

C. Godsil and G. Royle.Algebraic Graph Theory. Springer-Verlag, 2001

2001
[28]

M. Gromov. Asymptotic invariants of infinite groups. InGeometric group theory, Vol. 2 (Sussex, 1991), number 182 in London Math. Soc. Lecture Note Ser., pages 1–295. Camb. Univ. Press, 1993

1991
[29]

Gupta, I

A. Gupta, I. Newman, Y. Rabinovich, and A. Sinclair. Cuts, trees andℓ 1- embeddings of graphs.Combinatorica, 2:233–269, 2004

2004
[30]

Kleinberg and E

J. Kleinberg and E. Tardos.Algorithm Design. Pearson, 2005

2005
[31]

J. R. Lee. Separators in region intersection graphs. In C. H. Papadimitriou, editor, Proc. 8th Innovations in Theoretical Computer Science, volume 67 ofLIPIcs, pages 1:1–1:8. Schloss Dagstuhl, 2017

2017
[32]

C.-H. Liu. Assouad-Nagata dimension of minor-closed metrics. arXiv:2308.12273

arXiv
[33]

C.-H. Liu. Coarse Menger property of quasi-minor excluded graphs and length spaces. arXiv:2605.10068

arXiv
[34]

Lyndon and Paul E

Roger C. Lyndon and Paul E. Schupp.Combinatorial Group Theory. Springer Science & Business Media, January 2001

2001
[35]

J. M. Mackay, J. P. MacManus, and D. Spriano. Almost planar finitely presented groups. arXiv:2605.03040

arXiv
[36]

MacManus

J. MacManus. Accessibility, planar graphs, and quasi-isometries. arXiv:2310.15242

arXiv
[37]

MacManus

J. MacManus. Fat minors in finitely presented groups.Combinatorica, 45(40), 2025. 26

2025
[38]

Nguyen, A

T. Nguyen, A. Scott, and P. Seymour. Asymptotic structure. I. Coarse tree-width. arXiv:2501.09839

arXiv
[39]

Nguyen, A

T. Nguyen, A. Scott, and P. Seymour. Asymptotic structure. II. Path-width and additive quasi-isometry. Preprint 2024

2024
[40]

Nguyen, A

T. Nguyen, A. Scott, and P. Seymour. Asymptotic structure. VI. Distant paths across a disc. ArXiv:2509.07174. 27

arXiv

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [1]

Abrishami, M

T. Abrishami, M. Briański, J. Davies, X. Du, J. Masaříková, P. Rzążewski, and B. Walczak. Burling graphs in graphs with large chromatic number. InProceedings of the 2026 Annual ACM-SIAM Symposium on Discrete Algorithms (SODA), pages 3978–3998, 2026

work page 2026

[2] [2]

Albrechtsen, R

S. Albrechtsen, R. Diestel, A.-K. Elm, E. Fluck, R. W. Jacobs, P. Knappe, and P. Wollan. A structural duality for path-decompositions into parts of small radius. Innovations in Graph Theory, 3:207–246, 2026

2026

[3] [3]

Albrechtsen, M

S. Albrechtsen, M. Distel, and A. Georgakopoulos. ExcludingK2,t as a fat minor. arXiv:2510.14644

arXiv

[4] [4]

Albrechtsen, M

S. Albrechtsen, M. Distel, and A. Georgakopoulos. Small counterexamples to the fat minor conjecture. arXiv:2601.05761

arXiv

[5] [5]

A coarse block-cutvertex tree-decomposition

S. Albrechtsen and A. Georgakopoulos. A coarse block-cutvertex tree- decomposition. arXiv:2607.07030

work page Pith review arXiv

[6] [6]

Albrechtsen, R

S. Albrechtsen, R. Jacobs, P. Knappe, and P. Wollan. A characterisation of graphs quasi-isometric toK 4-minor-free graphs.Combinatorica, 45(61), 2025

work page 2025

[7] [7]

Berger and P

E. Berger and P. Seymour. Bounded diameter tree-decompositions.Combinatorica, 44(1):659–674, 2024

work page 2024

[8] [8]

Blazej, M

V. Blazej, M. Pilipczuk, and E. Protopapas. A coarse Menger’s Theorem for planar and bounded genus graphs. arXiv:2605.11112

arXiv