Tree-width of a graph excluding an apex-forest or a wheel as a minor

Chun-Hung Liu; Youngho Yoo

REVIEW 1 major objections 3 minor 30 references

Tree-width of a graph excluding an apex-forest or a wheel as a minor

T0 review · 1 major / 3 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read Every apex-forest H is forced as a minor once tree-width reaches |V(H)|-1, and wheels obey a near-linear bound.

desk verdict Genuinely new results worth refereeing, but Lemma 3.4 has a real contraction/lifting gap that the authors need to fix; the reader's own thin-wrist concern is not valid. read the letter →

arxiv 2509.09895 v1 pith:66OQDCES submitted 2025-09-11 math.CO

Chun-Hung Liu , Youngho Yoo This is my paper · ORCID

classification math.CO MSC 05C8305C75

keywords tree-widthgraphminorsapex-forestwheelGridMinorTheoremtree-decompositionthinoctopusminor-freegraphs

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

The paper studies the smallest tree-width f(H) that forces a planar graph H to appear as a minor. It proves that for every apex-forest H, a graph that becomes a forest after deleting one vertex, this threshold is exactly |V(H)|-1, the smallest value it can possibly have. That means the complete graph on |V(H)|-1 vertices is the only obstruction: no graph with tree-width below the threshold contains H, and every graph at the threshold does. The paper also proves a near-linear upper bound for wheels, f(H) ≤ max(3/2|V(H)|-9/2, |V(H)|-1), with sharp equality when |V(H)|≤7. These results extend the known sharp cases for trees and cycles and improve the previous bounds for both families.

What carries the argument

The machinery for the apex-forest theorem is the $(S,w)$-octopus: an $S$-rooted tree-decomposition in which every bag of size at least $w+1$ is a non-root leaf whose parent bag has size at most $|S|$. A wrist is a node with a child bag of size at least $w+1$, and the octopus is thin when no wrist has bag size equal to $|S|$; the induction in Lemma 3.4 is designed to produce a thin $(S,|V(F)|)$-octopus whenever $G[S]$ contains a spanning subgraph isomorphic to a subtree of $F$. The wheel theorem uses a different mechanism: it isolates a largest component $M$ of $G-(V(C)\cup V(P'))$, forms the reduced graph $G^*=G[V(C)\cup A]$ with $A$ the attachment vertices on the path $P'$, and builds a four-bag path-decomposition of $G^*$ whose largest bag has size at most $\max\{\frac32 k-3,k\}$.

What would settle it

Run the construction of Lemma 3.4 on graphs satisfying its hypotheses and check whether the minimal nontrivial $(S,|V(F)|)$-octopus can have a wrist bag larger than $S$; if it can, the contradiction in Theorem 3.5 fails as written. A decisive counterexample to the theorem would be any graph with tree-width $|V(H)|-1$ that excludes an apex-forest $H$ as a minor.

Watch

Extended reading notes

Core claim

The central claim is that the trivial lower bound f(H) ≥ |V(H)|-1 is attained for every apex-forest H. The proof works by assuming a graph G excludes $F^+$ as a minor, where $F$ is a tree and $F^+$ adds one vertex adjacent to every vertex of $F$, and then constructs a rooted tree-decomposition of G in which every bag has size at most |V(F)|. The construction is inductive and maintains a thin octopus: a rooted decomposition where large bags are confined to leaves whose parent bags are small. At the end the octopus has no wrists at all, which forces the decomposition width below |V(F)|. For wheels, the paper shows the threshold is at most max(3/2|V(H)|-9/2, |V(H)|-1) by reducing to a small graph built from the rim cycle, a path, and a largest component, and then exhibiting a four-bag path-decomposition of that reduced graph with controlled bag sizes.

Load-bearing premise

The load-bearing premise is that the thin octopus built in Lemma 3.4 has no wrist whose bag is larger than the root bag $S$; the lemma as written only rules out wrist bags of size exactly $|S|$, so if a larger wrist bag can occur, the final step of Theorem 3.5 does not follow.

Editorial extensions

If this is right

Every apex-forest-minor-free graph is $(|V(H)|-2)$-degenerate, so it can be colored with $|V(H)|-1$ colors.
The apex-forest case adds infinitely many planar graphs $H$ for which the trivial lower bound $f(H) \ge |V(H)|-1$ is tight, joining trees, cycles, and the small exceptional graphs already known.
For every wheel $H$, every graph with tree-width at least $\max\{\frac32|V(H)|-\frac92, |V(H)|-1\}$ contains $H$ as a minor, and for $|V(H)|\le 7$ the threshold is exactly $|V(H)|-1$.
Combined with the classification of graphs with no two disjoint cycles, positive answers to the paper's three structural questions and an improvement of the wheel bound to $|V(H)|-1$ would imply $f(H)=|V(H)|-1$ for every planar graph with no two disjoint cycles.

Reading between the lines

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

An algorithmic consequence the authors leave implicit is that apex-forest-minor-free graphs admit a simple greedy coloring with $|V(H)|-1$ colors in linear time, because the degeneracy bound follows directly from the theorem.
The $3/2$ coefficient in the wheel theorem appears to be an artifact of the four-bag path-decomposition of the reduced graph $G^*$, not of any known lower bound; constructing a longer path-decomposition might yield $f(W_k)=k-1$ for all $k$, which the paper's small-wheel equality supports.
A direct next test is whether the wheel theorem's path-decomposition survives subdividing rim edges; that would address the paper's first structural question for wheels, which is one of the main missing cases between the apex-forest and wheel results.

Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

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

The apex-forest theorem is a real step forward, not just an incremental tweak. It generalizes the known tree and cycle cases to all apex-forests, hits the trivial lower bound, and improves the Leaf–Seymour constant. The wheel bound is also a solid improvement over previous results, and the small-wheel exact values are a nice bonus. I found no circularity and no signs of cooked evidence; the proofs are long but follow the established octopus/tree-decomposition toolkit. The wheel section in particular reads carefully and I could not find a hole in it.

The soft spot is in Lemma 3.4, and it is not the one the reader flagged. The reader's concern about thin wrists is actually a non-issue: in an (S,w)-octopus, if a node t is a wrist, then it has a child c with |X_c| >= w+1, so by the defining property of an octopus the parent bag satisfies |X_t| <= |S|. Thin just forbids equality, so thin really does give |X_t| < |S| for every wrist. That part of Theorem 3.5 is fine.

The real problem is the attach step in Lemma 3.4. The proof takes a thin (X_t, |V(F)|)-octopus of the contracted graph G'_{t,c}, whose vertex set is essentially X_t ∪ X_c because the paths P_t and the set S have been contracted into X_t. It then tries to attach those decompositions to a tree-decomposition of the original graph G. But the attach operation defined in Section 2 requires V(G) ∩ V(G_i) to be a bag of both decompositions; here G'_{t,c} is not a subgraph of G, and all the internal vertices of the paths P_t (and the vertices of S) are missing from the bags of T_{t,c}. To get a true tree-decomposition of G you need to uncontract the paths, replacing each vertex of X_t in the bags by the whole path. That lifting step is not written down, and it is not obvious that it preserves the octopus bounds: a bag that contains several vertices of X_t could balloon in size once the full paths are reinserted. The sentence "It is easy to see" hides exactly this difficulty. I would not be surprised if the argument can be repaired, but as written the proof of Lemma 3.4 does not go through.

Who is this for? Graph minors people and anyone working on Seymour's question or degeneracy of H-minor-free graphs. The paper deserves a serious referee, and I would send it to review, but I would make the authors state the lifting argument explicitly or replace it with a different induction. The reader's verdict of CONDITIONAL is about right, just for the wrong reason.

Referee Report

1 major / 3 minor

Summary. The paper studies the function f(H), the smallest integer such that every graph of tree-width at least f(H) contains the planar graph H as a minor, for two families of planar graphs with no two disjoint cycles. Theorem 1.1 asserts that f(H)=|V(H)|-1 for every apex-forest H, improving a bound of Leaf and Seymour and extending the known cases of forests and cycles. Theorem 1.2 asserts that for every wheel H, f(H) is at most max(3/2 |V(H)| - 9/2, |V(H)|-1), with equality when |V(H)| is at most 7. The proofs introduce rooted tree-decompositions called octopuses, which are used in an induction for apex-forests, and a separate induction for wheels. The wheel proof appears to be a complete structural induction, while the apex-forest proof has a significant missing step in the central lemma.

Significance. If the results are correct, they constitute a genuine advance: Theorem 1.1 is sharp for a broad class not previously known to satisfy the trivial lower bound, and the wheel bound improves substantially on earlier polynomial bounds. The octopus machinery is an interesting proof tool, and the explicit conjectures connecting these bounds to Lovász's characterization are useful for future work. The proofs rely on standard tools (Menger's theorem, tree-decomposition connectivity) rather than on unproved assumptions. However, the central apex-forest proof has a load-bearing gap in Lemma 3.4, so Theorem 1.1 is not established as written; the wheel part does not appear to have a comparable problem.

major comments (1)

[Section 3, Lemma 3.4 (final attachment step)] The induction hypothesis is applied to G'_{t,c}, the graph obtained from G_{t,c} by contracting each path in P_t into its endpoint in X^3_t. The resulting tree-decomposition (T_{t,c}, X_{t,c}) is therefore a decomposition of G'_{t,c}, not of G_{t,c}. Its bags contain no internal vertices of the paths P_t and no vertices of S that have been identified with X^3_t. The attaching operation defined in Section 2 requires each attached decomposition to be a decomposition of a graph whose vertex set meets the remaining graph in a single bag. Since V(G'_{t,c}) is a quotient of V(G_{t,c}) rather than a subset of V(G), attaching (T_{t,c}, X_{t,c}) to (T^3, X^3) along X^3_t does not yield a tree-decomposition of G. In particular, if a vertex of G lies on one of the contracted paths P_t and occurs only in a deleted bag X^3_c, it is absent from every bag of (T^*, X^*), so (T^*, X^*) is not shown to cover that vertex or the edges incident with it. A lifting, i.e. uncontracting, argument is needed that reinserts the vertices of P_t into bags while keeping the root bag exactly X^3_t and preserving the property of being a thin (X^3_t, |V(F)|)-octopus. Such an argument is not supplied, and it is not routine because replacing a contracted vertex by a path inside bags can create new bags larger than |V(F)|, potentially violating the octopus condition. This gap is load-bearing: without it, Lemma 3.4 does not imply Theorem 3.5 or Theorem 1.1.

minor comments (3)

[Section 3, Claim 3.4.1] In the equality case of Claim 3.4.1, the sentence claiming that restricting (T^3, X^3) to the two nodes t and c yields a non-trivial (S, |V(F)|)-octopus of G is not justified and appears false in general, since edges between S \ X^3_c and V(G) - S need not be covered by the two bags. The subsequent argument using the vertex u and the fact that all other bags are contained in S proves the required conclusion directly; the restriction sentence and the appeal to minimality of |V(T^3)| should be removed or replaced by this direct argument.
[Section 3, Theorem 3.5] The final step of Theorem 3.5 is correct, but it is worth spelling out explicitly: if c is a child of a wrist t and |X_c| is at least |V(F)|+1, then the octopus condition directly gives |X_t| ≤ |S| = 1, and thinness excludes equality, so |X_t| = 0. A reader could otherwise think that thinness alone does not exclude wrist bags larger than |S|.
[Section 1 and Section 3] The paper states several times that graphs in the sets F(H) from the literature have tree-width at most |V(H)|-2 but omits proofs of these claims. This is acceptable for context, but the omission should be stated more prominently so that readers do not confuse these auxiliary claims with the main theorems.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main theorems are derived by inductive tree-decomposition arguments from standard tools, and self-citations are contextual only.

full rationale

The derivation chain is self-contained. For Theorem 1.1, the paper reduces apex-forests to graphs F^+ with F a tree by adding edges, defines (S,w)-octopuses as a proof device, and proves Lemma 3.4 by lexicographic induction on (|V(F)| - |V(F')|, |V(G)|), using Lemma 3.3 (Menger's theorem) to obtain disjoint paths and Lemma 3.2 as a base case. The target statement f(H)=|V(H)|-1 is not an input to the octopus definition or to any lemma; it is derived as the conclusion that every bag has size at most |V(F)|. Known results for forests and cycles are used only as context or as special cases, not as premises. The self-citation [21] is mentioned as a consequence of Theorem 1.1 ('recovers a special case'), and [18] is cited only in the discussion of conjectures and applications, so no load-bearing step rests on the authors' own prior work. Theorem 1.2 similarly proceeds by induction on |V(G)| and |V(G)|-|V(C)| with explicit bag constructions; no parameter is fitted to a data subset and no prediction is equivalent by construction to an input. Therefore no circular step is present.

Assumptions & free parameters 0 free parameters · 3 assumptions · 0 invented entities

The central claim rests on standard graph-theoretic tools (Menger's theorem, tree-decomposition basics) and on the assumption of finite simple graphs. No parameters are fitted to data. The octopus is a definition, not an unexplained postulate. The only unstated load-bearing invariant is identified in red_flags.

assumptions (3)

standard math Menger's theorem, in the form that a smallest (A,B)-separator has size equal to the maximum number of disjoint A-B paths.
Invoked in the proof of Lemma 3.3 (Section 3) to convert the absence of |S| disjoint paths from S to X_z into a separation with |A∩B| ≤ |S|−1 and then to re-root the octopus.
standard math Standard properties of tree-decompositions, including the ability to attach decompositions along a bag and the connectivity condition for vertex sets.
Used throughout Sections 3 and 4 when constructing and attaching (S,w)-octopuses and V(C)-rooted decompositions.
domain assumption The graphs are finite and simple.
Stated in the introduction; the proofs assume simple finite graphs, which is standard for the area.

how reviews work

0 comments

Cite this review

Pith. "Pith review of Tree-width of a graph excluding an apex-forest or a wheel as a minor." pith.science (2026). https://pith.science/paper/66OQDCES

@misc{pith2026250909895,
  author       = {Pith},
  title        = {Pith review of: Tree-width of a graph excluding an apex-forest or a wheel as a minor},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/66OQDCES}},
  note         = {Machine review of arXiv:2509.09895}
}

abstract

The Grid Minor Theorem states that for every planar graph $H$, there exists a smallest integer $f(H)$ such that every graph with tree-width at least $f(H)$ contains $H$ as a minor. The only known lower bounds on $f(H)$ beyond the trivial bound $f(H)\geq |V(H)|-1$ come from the maximum number of disjoint cycles in $H$. In this paper, we study $f(H)$ for planar graphs $H$ with no two disjoint cycles. We prove that $f(H)=|V(H)|-1$ for every apex-forest $H$. This result improves a bound of Leaf and Seymour and contains all known large graphs $H$ meeting the trivial lower bound to our knowledge. We also prove that $f(H)\leq \max\{\tfrac32|V(H)|-\tfrac92,|V(H)|-1\}$ for every wheel $H$.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

30 extracted references · 28 canonical work pages

[1]

Stefan Arnborg, Andrzej Proskurowski, and Derek G. Corneil. Forbidden minors characterization of partial 3-trees.Discrete Mathematics, 80(1):1–19, 1990

1990
[2]

Seymour, and Robin Thomas

Daniel Bienstock, Neil Robertson, Paul D. Seymour, and Robin Thomas. Quickly excluding a forest.Journal of Combinatorial Theory, Series B, 52(2):274–283, 1991

work page 1991
[3]

Tree-width and circumference of graphs.Journal of Graph theory, 43(1):24–25, 2003

Etienne Birmele. Tree-width and circumference of graphs.Journal of Graph theory, 43(1):24–25, 2003

work page 2003
[4]

Bodlaender, Jan van Leeuwen, Richard Tan, and Dimitrios M

Hans L. Bodlaender, Jan van Leeuwen, Richard Tan, and Dimitrios M. Thi- likos. On interval routing schemes and treewidth.Information and Computation, 139(1):92–109, 1997

work page 1997
[5]

Dover Books on Mathematics

Bela Bollobas.Extremal Graph Theory. Dover Books on Mathematics. Dover Publications, Mineola, NY, June 2004

work page 2004
[6]

Polynomial bounds for the grid-minor the- orem.Journal of the ACM (JACM), 63(5):1–65, 2016

Chandra Chekuri and Julia Chuzhoy. Polynomial bounds for the grid-minor the- orem.Journal of the ACM (JACM), 63(5):1–65, 2016

work page 2016
[7]

Excluded grid theorem: Improved and simplified

Julia Chuzhoy. Excluded grid theorem: Improved and simplified. InProceedings of the forty-seventh annual ACM symposium on Theory of Computing, pages 645– 654, 2015

work page 2015
[8]

Improved bounds for the excluded grid theorem.arXiv preprint arXiv:1602.02629, 2016

Julia Chuzhoy. Improved bounds for the excluded grid theorem.arXiv preprint arXiv:1602.02629, 2016

arXiv 2016

Show all 30 references

[9]

Towards tight(er) bounds for the excluded grid theorem.Journal of Combinatorial Theory, Series B, 146:219–265, 2021

Julia Chuzhoy and Zihan Tan. Towards tight(er) bounds for the excluded grid theorem.Journal of Combinatorial Theory, Series B, 146:219–265, 2021

2021
[10]

Youssou Dieng.D´ ecomposition arborescente des graphes planaires et routage com- pact. Ph.D. thesis, L’Universit´ e Bordeaux I, 2009

2009
[11]

Graph minors I: A short proof of the path-width theorem

Reinhard Diestel. Graph minors I: A short proof of the path-width theorem. Combinatorics, Probability and Computing, 4(1):27–30, 1995

1995
[12]

A characterization of graphs with no octahedron minor.Journal of Graph Theory, 74(2):143–162, 2013

Guoli Ding. A characterization of graphs with no octahedron minor.Journal of Graph Theory, 74(2):143–162, 2013

2013
[13]

A chain theorem for 3 +-connected graphs.SIAM Journal on Discrete Mathematics, 26(1):102–113, 2012

Guoli Ding and Cheng Liu. A chain theorem for 3 +-connected graphs.SIAM Journal on Discrete Mathematics, 26(1):102–113, 2012

2012
[14]

Excluding a small minor.Discrete Applied Mathe- matics, 161(3):355–368, 2013

Guoli Ding and Cheng Liu. Excluding a small minor.Discrete Applied Mathe- matics, 161(3):355–368, 2013

2013
[15]

Gabriel A. Dirac. A property of 4-chromatic graphs and some remarks on critical graphs.Journal of the London Mathematical Society, 1(1):85–92, 1952

1952
[16]

Fellows and Michael A

Michael R. Fellows and Michael A. Langston. On search decision and the efficiency of polynomial-time algorithms. InProceedings of the twenty-first annual ACM symposium on Theory of Computing, pages 501–512, 1989

1989
[17]

Pascal Gollin, Kevin Hendrey, Sang-il Oum, and Bruce Reed

J. Pascal Gollin, Kevin Hendrey, Sang-il Oum, and Bruce Reed. Linear bounds on treewidth in terms of excluded planar minors.arXiv preprint arXiv:2402.17255, 2024. 15

2024
[18]

Bounds on treewidth via excluding disjoint unions of cycles.arXiv preprint arXiv:2501.01703, 2025

Meike Hatzel, Chun-Hung Liu, Bruce Reed, and Sebastian Wiederrecht. Bounds on treewidth via excluding disjoint unions of cycles.arXiv preprint arXiv:2501.01703, 2025

2025 arXiv
[19]

Linear min-max relation between the treewidth of anH-minor-free graph and its largest grid minor.Journal of Combinatorial Theory, Series B, 141:165–180, 2020

Ken-ichi Kawarabayashi and Yusuke Kobayashi. Linear min-max relation between the treewidth of anH-minor-free graph and its largest grid minor.Journal of Combinatorial Theory, Series B, 141:165–180, 2020

2020
[20]

Tree-width and planar minors.Journal of Combinatorial Theory, Series B, 111:38–53, 2015

Alexander Leaf and Paul Seymour. Tree-width and planar minors.Journal of Combinatorial Theory, Series B, 111:38–53, 2015

2015
[21]

Tight minimum degree conditions for apex- outerplanar minors and subdivisions in graphs and digraphs.arXiv preprint arXiv:2403.11470, 2024

Chun-Hung Liu and Youngho Yoo. Tight minimum degree conditions for apex- outerplanar minors and subdivisions in graphs and digraphs.arXiv preprint arXiv:2403.11470, 2024

2024 arXiv
[22]

On graphs not containing independent circuits.Matematikai Lapok, 16(289-299):7, 1965

L´ aszl´ o Lov´ asz. On graphs not containing independent circuits.Matematikai Lapok, 16(289-299):7, 1965

1965
[23]

A characterization of graphs with no cube minor.Journal of Combinatorial Theory, Series B, 80(2):179–201, 2000

John Maharry. A characterization of graphs with no cube minor.Journal of Combinatorial Theory, Series B, 80(2):179–201, 2000

2000
[24]

Thilikos

Jean-Florent Raymond and Dimitrios M. Thilikos. Low polynomial exclusion of planar graph patterns.Journal of Graph Theory, 84(1):26–44, 2017

2017
[25]

Quickly excluding a planar graph.Journal of Combinatorial Theory, Series B, 62(2):323–348, 1994

Neil Robertson, Paul Seymour, and Robin Thomas. Quickly excluding a planar graph.Journal of Combinatorial Theory, Series B, 62(2):323–348, 1994

1994
[26]

Neil Robertson and Paul D. Seymour. Graph minors. X. Obstructions to tree- decomposition.Journal of Combinatorial Theory, Series B, 52(2):153–190, 1991

1991
[27]

Satyanarayana and L

A. Satyanarayana and L. Tung. A characterization of partial 3-trees.Networks, 20(3):299–322, 1990

1990
[28]

Hadwiger’s conjecture.Open problems in mathematics, pages 417–437, 2016

Paul Seymour. Hadwiger’s conjecture.Open problems in mathematics, pages 417–437, 2016

2016
[29]

A shorter proof of the path-width theorem.arXiv preprint arXiv:2309.05100, 2023

Paul Seymour. A shorter proof of the path-width theorem.arXiv preprint arXiv:2309.05100, 2023

2023 arXiv
[30]

Bemerkungen zu Hadwigers Vermutung.Mathematische Annalen, 141:433–451, 1960

Klaus Wagner. Bemerkungen zu Hadwigers Vermutung.Mathematische Annalen, 141:433–451, 1960. 16

1960

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.

[1] [1]

Stefan Arnborg, Andrzej Proskurowski, and Derek G. Corneil. Forbidden minors characterization of partial 3-trees.Discrete Mathematics, 80(1):1–19, 1990

1990

[2] [2]

Seymour, and Robin Thomas

Daniel Bienstock, Neil Robertson, Paul D. Seymour, and Robin Thomas. Quickly excluding a forest.Journal of Combinatorial Theory, Series B, 52(2):274–283, 1991

work page 1991

[3] [3]

Tree-width and circumference of graphs.Journal of Graph theory, 43(1):24–25, 2003

Etienne Birmele. Tree-width and circumference of graphs.Journal of Graph theory, 43(1):24–25, 2003

work page 2003

[4] [4]

Bodlaender, Jan van Leeuwen, Richard Tan, and Dimitrios M

Hans L. Bodlaender, Jan van Leeuwen, Richard Tan, and Dimitrios M. Thi- likos. On interval routing schemes and treewidth.Information and Computation, 139(1):92–109, 1997

work page 1997

[5] [5]

Dover Books on Mathematics

Bela Bollobas.Extremal Graph Theory. Dover Books on Mathematics. Dover Publications, Mineola, NY, June 2004

work page 2004

[6] [6]

Polynomial bounds for the grid-minor the- orem.Journal of the ACM (JACM), 63(5):1–65, 2016

Chandra Chekuri and Julia Chuzhoy. Polynomial bounds for the grid-minor the- orem.Journal of the ACM (JACM), 63(5):1–65, 2016

work page 2016

[7] [7]

Excluded grid theorem: Improved and simplified

Julia Chuzhoy. Excluded grid theorem: Improved and simplified. InProceedings of the forty-seventh annual ACM symposium on Theory of Computing, pages 645– 654, 2015

work page 2015

[8] [8]

Improved bounds for the excluded grid theorem.arXiv preprint arXiv:1602.02629, 2016

Julia Chuzhoy. Improved bounds for the excluded grid theorem.arXiv preprint arXiv:1602.02629, 2016

arXiv 2016