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The paper proves that in K_{1,d}-free graphs, excluding the k-ladder as an induced minor forces a tree-decomposition whose bags have bounded independence number.

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2026-08-05 10:27 UTC pith:ZVMGMHBM

load-bearing objection Genuine new width-obstruction result for ladders in K_{1,d}-free graphs, with coherent proof and useful corollaries; the only real blemish is an unproved algorithmic claim in Section 1.1. the 2 major comments →

arxiv 2509.04026 v1 pith:ZVMGMHBM submitted 2025-09-04 math.CO cs.DM

Excluding a Ladder as an Induced Minor in Graphs Without Induced Stars

Mujin Choi , Sebastian Wiederrecht This is my paper
classification math.CO cs.DM MSC 05C7505C8305C69
keywords induced minorsladder graphstree-independence numberK_{1,d}-free graphsstrong bramblesinduced Erdos-Posa propertylong thetaslong prisms
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

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

The paper's goal is a structural dichotomy: any graph that avoids a large induced star and also avoids a k-ladder as an induced minor must have small tree-independence number. Equivalently, if tree-independence number is large, the graph must contain a k-skinny ladder (a ladder with each rung subdivided once) as an induced minor. The proof builds the ladder by finding two non-adjacent induced paths and many mutually non-adjacent connector paths, then sorting them into a rope ladder that contracts to the skinny ladder. This is the first step toward an induced grid theorem for K_{1,d}-free graphs, and it implies several known results about wheels, long thetas, long prisms, and induced cycles.

Core claim

The central claim is Theorem 1.4: there is a function tau(k,d) such that every K_{1,d}-free graph with tree-independence number at least tau(k,d) contains the k-skinny ladder as an induced minor. Since the skinny ladder contains the k-ladder as an induced minor, the analogous statement for k-ladders follows. The proof proceeds from a large tree-independence number to a strong bramble, then to two non-adjacent induced paths whose closed neighbourhoods cannot be separated by a small-independence set, then to a large shuffled rope ladder, then to a rope ladder, and finally to the skinny ladder by contraction.

What carries the argument

Shuffled rope ladders: two non-adjacent induced paths P1,P2 plus k pairwise non-adjacent induced paths Phi_i, each with neighbours on both rails. A cleaning lemma sorts the attachment orders to make a rope ladder; contracting its rungs gives the k-skinny ladder. The engine is Theorem 6.1, a Menger-like statement that turns a separator-resistance condition between a fixed subgraph H and an induced path P into an ell-H-rope ladder. Its proof greedily selects shortest mutually non-adjacent paths while managing endpoints and second-endpoints, which keeps the final rail path induced.

Load-bearing premise

The proof of Theorem 6.1 relies on each chosen connector path attaching to the previous path through a vertex that is not an endpoint or second-endpoint; if attachments could only happen at those boundary vertices, the assembled rail path would not be induced.

What would settle it

Build a K_{1,3}-free graph with two non-adjacent induced paths P and H such that every separator between N[P] and N[H] has independence number at least eta(2,3)=432, yet no 2-H-rope ladder appears as an induced subgraph.

Watch this falsifier. Get emailed when new claim-graph text bears on it.

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If this is right

  • In K_{1,d}-free graphs, forbidding the k-ladder as induced minor bounds the independence number of every bag in a tree-decomposition.
  • The skinny-ladder version yields an independence Erdos-Posa property for every connected induced subgraph of a k-skinny ladder.
  • The method gives a sharper bound for wheel exclusion and a generalisation of the theta/prism theorem to k-long thetas and prisms.
  • The proof is constructive, so for fixed k,d one can either find the ladder or construct the decomposition algorithmically.
  • The same construction underlies several previously separate results, suggesting the ladder is the common obstruction.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • Theorem 6.1 fixes one side H and allows the path P to be replaced; a symmetric, two-side-fixed version would amount to an induced Menger theorem that remains open.
  • The bounds are recursive and likely far from optimal, so the theorem should be read as qualitative.
  • If one could force rung paths to be distance-2 separated, the double-ladder step toward the induced grid would become visible.
  • The cycle-rope-ladder structure could be the basis for a two-cycle version needed for double wheels.

Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper proves that for every positive integer k and d≥2, every K_{1,d}-free graph either contains the k-skinny ladder as an induced minor or has tree-independence number bounded by an explicit function τ(k,d). Since the k-skinny ladder contains the k-ladder as an induced minor, this establishes the headline ladder theorem. The proof combines a strong-bramble duality theorem (Theorem 2.3), a lemma producing two non-adjacent induced paths whose closed neighborhoods cannot be separated by a set of small independence number (Theorem 5.2), and a Menger-like construction (Theorem 6.1) that builds a large shuffled rope ladder, which is then cleaned into a rope ladder (Lemma 4.3) and converted into a skinny ladder. The paper also derives applications: an improved bound for wheel exclusion, a bounded tree-independence number for graphs excluding long thetas and long prisms, and an induced Erdos-Posa-type statement for connected induced subgraphs of skinny ladders in K_{1,d}-free graphs.

Significance. If correct, the main theorem is a substantial advance toward the induced grid conjecture for K_{1,d}-free graphs: it gives the first family of planar obstructions handled beyond wheels/tripods, with explicit elementary bounds and a relatively uniform proof strategy. The paper is honest about using strong bramble duality and the prior result Lemma 2.4 from [CHMW25] as black boxes, and these uses are legitimate. The applications to wheels, long thetas/prisms, and induced Erdos-Posa are nontrivial and improve known bounds. The main proof chain from Theorem 5.2, Theorem 6.1, and Lemma 4.3 to Theorem 1.4 is internally consistent, and I found no circularity. The manuscript would be a valuable contribution after addressing the unproved algorithmic claim and a few local issues in the applications.

major comments (2)
  1. [§1.1] The paper states: 'our proofs are constructive, and with the result of [CHMW25], they imply an algorithm ... in time |V(G)|^{g(k,d)}'. No algorithm or complexity analysis is provided anywhere in the manuscript. Since this is advertised as a contribution, please either give a precise constructive argument (including the choice of g) or remove/qualify the claim. This does not affect the correctness of Theorem 1.4, but as written it is an unsupported statement.
  2. [Lemma 7.10] The statement assumes ℓ ≥ 12(d−1)^2(2k−1)+1, but the greedy argument in the proof discards up to 4(d−1)^2(2k−1) indices after each chosen index and needs four chosen indices. The proof itself later uses ℓ ≥ 16(d−1)^2(2k−1). With the stated 12(d−1)^2(2k−1)+1 the greedy selection may fail. This is a local numerical error, but it affects Theorem 1.6 and should be corrected in the statement and in Corollary 7.7.
minor comments (5)
  1. [§6.1] The final step of Theorem 6.1 asserts that each R_i has length at least 4 'since w_i is adjacent to a vertex in R_i that is not an endpoint or second-endpoint'. This is correct, but the one-sentence justification relies on the earlier deletion of S^i and on the definition of Qbar^i_{t_i}; please expand it for readability.
  2. [§6.1] In the induction step, 'ℓi' should be 'ℓ^i' (powers of ℓ), matching the later bounds involving ℓ^j. The current notation could confuse.
  3. [§7.1] In the proof of Theorem 7.3, 'β(s´2)' appears to be a typo for 'β(s_2)'. Please correct.
  4. [Throughout] There are several typos: 'grpah' (p.1), 'combarability' (p.1), 'T heorem1.4' (p.3), 'InrCHT24s' (p.21), 'doe snot' (p.19), 'we loose control' (p.25). A careful proofreading pass is needed.
  5. [Theorem 5.2] Theorem 5.2 is stated as a direct consequence of the proof of Theorem 5.1, but it is not formally proved. Since it is load-bearing for Theorem 1.4, please include a short proof or explicitly say it follows by the same argument with P2 = R.

Circularity Check

0 steps flagged

No significant circularity: Theorem 1.4 is derived from bramble duality and independent cleaning/construction lemmas; the only caveat is an unproved algorithmic claim in Section 1.1.

full rationale

The derivation chain for the main theorem is non-circular. Theorem 1.4 is obtained by combining Theorem 5.2 (which produces two non-adjacent induced paths whose closed neighborhoods cannot be separated by small-independence sets), Theorem 6.1 (which converts such a pair into a large shuffled rope ladder), and Lemma 4.3 (which cleans a shuffled rope ladder into a rope ladder, and hence into a skinny ladder as an induced minor). None of these intermediate statements assumes the excluded-ladder conclusion. The bramble-duality Theorem 2.3 is attributed to Adler and to the authors' prior paper [CHMW25], but it is a parameter-free structural statement about tree-independence number and strong brambles, not about ladders. Lemma 2.4 from [CHMW25] is also a self-citation, but it is a separate, parameter-free lemma about finding an induced path meeting every bramble element; it does not state or assume Theorem 1.4, and it is used as a black-box tool rather than as a renamed version of the target result. The flagged assertion in Theorem 6.1 that each selected path R_i has length at least 4 is justified by the construction: w_{i+1} belongs to Qbar^i_{t_i} = N[R_i] \setminus N[S^i], so its neighbor in R_i is not an endpoint or second-endpoint, forcing at least four vertices on R_i. The only unsupported statement found is in Section 1.1, where the paper claims that the constructive proofs imply an algorithm running in time |V(G)|^{g(k,d)} without giving an algorithm or complexity analysis. This is a missing proof/correctness-risk, not circularity; it does not make the main theorem reduce to its inputs. Accordingly, no circular step is identified and the score is 0.

Axiom & Free-Parameter Ledger

4 free parameters · 4 axioms · 0 invented entities

The paper introduces rope-ladder families as proof-internal graph definitions, not as unexplained posited objects. All constants are hand-chosen to make constructions terminate. The main external inputs are bramble duality and the prior bramble-path lemma, which are parameter-free and do not assume the target theorem.

free parameters (4)
  • eta(ell,d)=8(d-1)ell^(ell+1) = 8(d-1)ell^(ell+1)
    Hand-chosen threshold in Theorem 6.1, large enough that the iterative construction can run ell+1 times while alpha(N^i) stays positive. Not fitted to data; a proof-design constant.
  • rho(k,d)=(2(d-1)^2(2d-1)^(2d-2))^(k-1) = (2(d-1)^2(2d-1)^(2d-2))^(k-1)
    Chosen in Lemma 4.1 so the induction on k survives the gap case distinction.
  • nu(k,d)=rho(rho((k-1)^2+1,d),d) = rho(rho((k-1)^2+1,d),d)
    Composed threshold in Lemma 4.3 so two applications of Lemma 4.1 leave enough rung paths for the Erdos-Szekeres sorting step.
  • tau(k,d)=8*eta(nu(k,d),d)+8d-14 = 8*eta(nu(k,d),d)+8d-14
    Final bound in Theorem 1.4, assembled directly from Theorem 5.2 and Theorem 6.1.
axioms (4)
  • standard math All graphs are finite and simple.
    Stated in Section 2 as the basic setting of the paper.
  • domain assumption The ambient restriction is K_{1,d}-freeness.
    All main theorems quantify over K_{1,d}-free graphs, and Observation 2.1 (at most 2(d-1) neighbors on an induced path or long cycle) is used throughout.
  • standard math Strong bramble duality and the bramble-path lemma (Theorem 2.3 and Lemma 2.4 from [CHMW25]).
    Borrowed prior results used in Section 5 to locate the initial induced path and cycle, converting large alpha-treewidth into separator lower bounds.
  • standard math Erdos-Szekeres theorem [ES35].
    Used in Lemma 4.3 to sort the rung paths into a monotone order.

pith-pipeline@v1.4.0-alltime-deepseek-medium · 26280 in / 18805 out tokens · 174572 ms · 2026-08-05T10:27:31.075723+00:00 · methodology

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Cite this review

Pith. "Pith review of Excluding a Ladder as an Induced Minor in Graphs Without Induced Stars." pith.science (2026). https://pith.science/paper/ZVMGMHBM

@misc{pith2026250904026,
  author       = {Pith},
  title        = {Pith review of: Excluding a Ladder as an Induced Minor in Graphs Without Induced Stars},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/ZVMGMHBM}},
  note         = {Machine review of arXiv:2509.04026}
}
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read the original abstract

A $k$-ladder is the graph obtained from two disjoint paths, each with $k$ vertices, by joining the $i$th vertices of both paths with an edge for each $i\in\{ 1,\ldots,k\}$. In this paper, we show that for all positive integers $k$ and $d$, the class of all $K_{1,d}$-free graphs excluding the $k$-ladder as an induced minor has a bounded tree-independence number. We further show that our method implies a number of known results: We improve the bound on the tree-independence number for the class of $K_{1,d}$-free graphs not containing a wheel as an induced minor given by Choi, Hilaire, Milani\v{c}, and Wiederrecht. Furthermore, we show that the class of $K_{1,d}$-free graphs not containing a theta or a prism, whose paths have length at least $k$, as an induced subgraph has bounded tree-independence number. This improves a result by Chudnovsky, Hajebi, and Trotignon. Finally, we extend the induced Erd\H{o}s-P\'osa result of Ahn, Gollin, Huynh, and Kwon in $K_{1,d}$-free graphs from long induced cycles to any graph that is an induced minor of the $k$-ladder where every edge is subdivided exactly once.

Figures

Figures reproduced from arXiv: 2509.04026 by Mujin Choi, Sebastian Wiederrecht.

Figure 1
Figure 1. Figure 1: A 4-ladder (left) and a 4-skinny ladder (right) [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: A k-long theta (left) and a k-long prism (right). Dashed lines represent paths of length at least k. The strengthening from ladders to skinny ladders allows us to deduce simple and unified proofs for several known results in the area as discussed below. In [CHMW25], the authors proved that every K1,d-free graph excluding a wheel as an induced minor has bounded tree-independence number where the bound is gi… view at source ↗
Figure 3
Figure 3. Figure 3: A 4-shuffled rope ladder (left) and a 4-rope ladder (right). [PITH_FULL_IMAGE:figures/full_fig_p009_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: A 4-cycle rope ladder. If k “ 1, we can choose B “ A and Q “ P. Now assume that k ě 2 and that the statement holds for k ´ 1. Since G is K1,d-free, each vertex in P is adjacent to at most d ´ 1 vertices in A. Hence, we may find A1 Ď A such that • a1 P A1 , • |A1 | ě |A|{p2pd ´ 1q 2 q ě p2d ´ 1q 2d´2ρpk ´ 1, dq, and • for each a, a1 P A1 , Npaq X Npa 1 q “ H. We may rename the vertices in A1 so that v i 1 a… view at source ↗
Figure 5
Figure 5. Figure 5: The two cases in the proof of Lemma 4.1; Cases 1 (Left): There are many vertices in A1 that have no neighbor before v 1 j but many of them have a neighbor before v 1 j`1 . In this case, we discard v 1 j`1 and all vertices after it. Case 2 (Right): There are many vertices in A1 that have no neighbor before v 1 degpa1q . Let A2 :“ pA1 zta1uq X pNp˚v 1 jP˚v 1 j`1 qzNp˚v 1 1P˚v 1 j qq be the set of all members… view at source ↗
Figure 6
Figure 6. Figure 6: After ℓ ` 1 steps, we can find a rail path P 1 (red) and rung paths Φi ’s (blue). For i P rℓs, among the vertices in Npw i`1 q X Ri , let wq i be the vertex farthest from H (in Ri ). If |Npw i`1 qXRi | “ 1, let ϕ i 2 be the neighbor of wq i in the path Ri closer to H (in Ri ). Otherwise, let ϕ i 2 be the vertex in Npw i`1 qXRi that is nearest to H (in Ri ). Let P 1 :“ wq 1w 2R2wq 2w 3R3wq 3 ¨ ¨ ¨ w ℓRℓwq ℓ… view at source ↗
Figure 7
Figure 7. Figure 7: From the left: a junction connecting P to Q of type 1, type 2, type 3, and the cleaning of a junction of type 3. 2. If |NpPqXV pQq| “ 2 and two vertices in NpPqXV pQq are adjacent, then we say the junction is type 2. 3. Otherwise, we say that the junction is type 3. Suppose that the junction connecting P to Q is type 3 where P and Q are induced paths and V pPq X NpQq “ tvu. Let w1, w2 P Npvq X P be the ver… view at source ↗
Figure 8
Figure 8. Figure 8: The 6 cases towards finding a k-long theta or a k-long prism in a cycle rope ladder in the proof of Lemma 7.10. Case 1: For each i P r4s, there is a unique junction area of Φi , say Di . Consider the type of junctions between Φi and Di . As there are three types of junctions and |A| “ 4, there must be i ‰ j such that Φi and Di has the same type of junction as Φj and Dj . For any i P rAs we refer to the typ… view at source ↗
Figure 9
Figure 9. Figure 9: Possible next steps: a double ladder (left) and a double wheel (right) [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗

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