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arxiv: 2510.15074 · v4 · submitted 2025-10-16 · 🧮 math.CO

(Treewidth, Clique)-Boundedness and Poly-logarithmic Tree-Independence

Maria Chudnovsky , Ajaykrishnan E S , Daniel Lokshtanov This is my paper

Pith reviewed 2026-05-18 05:59 UTC · model grok-4.3

classification 🧮 math.CO
keywords tree-independence numbertreewidthindependence containersgraph algorithmsDallard-Milanič-Štorgel conjecturebounded clique numberpoly-logarithmic bounds
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The pith

A slight variant of the Dallard-Milanič-Štorgel conjecture holds and preserves its algorithmic consequences for graphs.

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

The paper establishes that a modified form of a conjecture relating tree-independence number to treewidth remains valid. This variant keeps the key algorithmic advantages that motivated the original conjecture even though the full conjecture was recently disproved. The authors introduce independence-containers as a generalization of maximal cliques to help prove the result. A sympathetic reader would care because bounded tree-independence number yields efficient algorithms for hard problems such as independent set on restricted graph classes.

Core claim

We prove a slight variant of the Dallard-Milanič-Štorgel conjecture connecting tree-independence number to the classical notion of treewidth. This variant retains much of the algorithmic significance of the original statement. As part of the proof we introduce the notion of independence-containers, which can be viewed as a generalization of the set of all maximal cliques of a graph and is of independent interest.

What carries the argument

Independence-containers, a generalization of the set of all maximal cliques that supports bounding independent sets in tree-decomposition bags.

If this is right

  • Algorithmic results that previously relied on the original conjecture continue to hold under the variant.
  • Graphs with bounded treewidth and bounded clique number admit tree decompositions whose bags have independent-set size bounded by a poly-logarithmic function.
  • The new independence-container concept supplies a tool for deriving further bounds on tree-independence number.

Where Pith is reading between the lines

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

  • The result may extend to other width parameters that combine treewidth with clique constraints.
  • Independence-containers could be applied directly to obtain approximation algorithms for independent-set problems without full tree decompositions.
  • The approach suggests testing whether similar mild variants rescue other recently disproved width conjectures.

Load-bearing premise

The particular modification made to the original Dallard-Milanič-Štorgel conjecture is close enough to preserve the stated algorithmic consequences.

What would settle it

A graph class with bounded treewidth and clique number whose minimum tree-independence number grows faster than any poly-logarithmic function of the input size.

read the original abstract

An independent set in a graph $G$ is a set of pairwise non-adjacent vertices. A tree decomposition of $G$ is a pair $(T, \chi)$ where $T$ is a tree and $\chi : V(T) \rightarrow 2^{V(G)}$ is a function satisfying the following two axioms: for every edge $uv \in V(G)$ there is a $x \in V(T)$ such that $\{u,v\} \subseteq \chi(x)$, and for every vertex $u \in V(G)$ the set $\{x \in V(T) ~:~ u \in \chi(X)\}$ induces a non-empty and connected subtree of $T$. The sets $\chi(x)$ for $x \in V(T)$ are called the bags of the tree decomposition. The tree-independence number of $G$ is the minimum taken over all tree decompositions of $G$ of the maximum size of an independent set of the graph induced by a bag of the tree decomposition. The study of graph classes with bounded tree-independence number has attracted much attention in recent years, in part due its improtant algorithmic implications. A conjecture of Dallard, Milani\v{c} and \v{S}torgel, connecting tree-independence number to the classical notion of treewidth, was one of the motivating problems in the area. This conjecture was recently disproved, but here we prove a slight variant of it, that retains much of the algorithmic significance. As part of the proof we introduce the notion of independence-containers, which can be viewed as a generalization of the set of all maximal cliques of a graph, and is of independent interest.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 0 minor

Summary. The paper proves a slight variant of the disproved Dallard-Milanič-Štorgel conjecture, showing that (treewidth, clique)-bounded graph classes admit poly-logarithmic tree-independence number. It introduces independence-containers as a generalization of maximal cliques and uses them to establish the variant while claiming retention of much of the original conjecture's algorithmic significance.

Significance. If the variant is correctly formulated and the proof holds, the result would provide a usable substitute for the original conjecture, enabling algorithmic applications such as FPT algorithms or poly-logarithmic approximations for independent set and related problems in (tw, ω)-bounded classes via tree decompositions with small independent sets in bags.

major comments (2)
  1. The manuscript must explicitly state the precise formulation of the slight variant of the Dallard-Milanič-Štorgel conjecture (currently only alluded to in the abstract) and include a reduction or meta-theorem showing that the new bound on tree-independence number implies the same algorithmic corollaries (e.g., FPT or approximation results) that would have followed from the original conjecture. This comparison is load-bearing for the central claim that the variant 'retains much of the algorithmic significance.'
  2. The construction and properties of independence-containers (introduced as a generalization of maximal cliques) require a detailed verification that they suffice for the tree-decomposition arguments in the variant; without an explicit check that the algorithmic reductions from bounded tree-independence survive the change, the claim remains unanchored.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We agree that additional explicit statements and verifications will strengthen the presentation and will incorporate them in the revised version.

read point-by-point responses

  1. Referee: The manuscript must explicitly state the precise formulation of the slight variant of the Dallard-Milanič-Štorgel conjecture (currently only alluded to in the abstract) and include a reduction or meta-theorem showing that the new bound on tree-independence number implies the same algorithmic corollaries (e.g., FPT or approximation results) that would have followed from the original conjecture. This comparison is load-bearing for the central claim that the variant 'retains much of the algorithmic significance.'

    Authors: We agree that the precise formulation of the variant should be stated explicitly in the introduction. The variant of the Dallard-Milanič-Štorgel conjecture proved in the paper asserts that every (treewidth, clique)-bounded graph class has poly-logarithmic tree-independence number. In the revision we will add a meta-theorem (new Section 5) showing that a poly-logarithmic bound on tree-independence number yields the same FPT algorithms and poly-logarithmic approximations for independent set and related problems that would have followed from a bounded tree-independence number, with only a mild worsening of the running time from polynomial to n^{O(log n)}. revision: yes

  2. Referee: The construction and properties of independence-containers (introduced as a generalization of maximal cliques) require a detailed verification that they suffice for the tree-decomposition arguments in the variant; without an explicit check that the algorithmic reductions from bounded tree-independence survive the change, the claim remains unanchored.

    Authors: We thank the referee for this suggestion. Independence-containers are introduced in Section 3 as a generalization of the family of maximal cliques and are used to build the tree decompositions whose bags have poly-logarithmic independence number. In the revised manuscript we will expand Section 3 with a new subsection that verifies in detail the required properties of independence-containers and explicitly confirms that the standard algorithmic reductions from bounded tree-independence number carry over to the poly-logarithmic case, thereby anchoring the algorithmic claims. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The manuscript introduces independence-containers as a new combinatorial object generalizing maximal cliques and uses them to establish a slight variant of the Dallard-Milanič-Štorgel conjecture. The derivation proceeds via direct proof from standard tree-decomposition axioms and the newly defined containers rather than any self-referential definition, fitted parameter renamed as prediction, or load-bearing self-citation chain. The abstract explicitly frames the result as an independent proof that retains algorithmic consequences without reducing the claimed bound to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

Relies on standard definitions of tree decompositions, independent sets, and treewidth from prior graph theory literature; introduces independence-containers as a new tool for the proof.

axioms (1)
  • standard math Standard axioms for tree decompositions: every edge covered by some bag and vertex appearances form connected subtrees.
    Invoked in the definition of tree-independence number.
invented entities (1)
  • independence-containers no independent evidence
    purpose: Generalization of the set of all maximal cliques used to prove the poly-log bound.
    New object introduced in the paper; no independent evidence outside the proof is mentioned.

pith-pipeline@v0.9.0 · 5854 in / 1187 out tokens · 28539 ms · 2026-05-18T05:59:48.056345+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Tree-alpha and excluding finitely many graphs

    math.CO 2026-05 unverdicted novelty 8.0

    Hereditary classes defined by finitely many excluded induced subgraphs have bounded tree-α iff they are (tw,ω)-bounded, i.e., exclude K_{a,a}, forests with components of at most three leaves, and their line graphs.

Reference graph

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