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arxiv: 2303.13655 · v4 · pith:2YRKWGU6new · submitted 2023-03-23 · 🧮 math.CO · cs.DM

Clustered independence and bounded treewidth

Kolja Knauer , Torsten Ueckerdt This is my paper

Pith reviewed 2026-05-24 09:51 UTC · model grok-4.3

classification 🧮 math.CO cs.DM
keywords clustered independencetreewidthc-clustered setindependence numbergraph partitioninglower boundstight constructions
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The pith

Graphs of treewidth k always contain a c-clustered set of size at least c n over (c plus k plus one).

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

The paper proves a lower bound on the largest c-clustered set in any graph of treewidth k. A c-clustered set is any vertex subset whose induced subgraph has all components of size at most c. The bound states that such a set must have size at least c n divided by (c plus k plus one). It improves an earlier result and settles a conjecture from 2013 in the cases c at most 2 or k equal to 1. A reader cares because the result gives a guaranteed way to partition low-treewidth graphs into small induced pieces.

Core claim

For any graph G on n vertices of treewidth k, the c-clustered independence number satisfies alpha_c(G) at least c over (c plus k plus one) times n. When c is at most 2 or k equals 1 the bound strengthens to c over (c plus k) times n. When c equals 3 and k equals 2 the bound is 5 over 9 times n. Each of these lower bounds is matched by explicit constructions of n-vertex graphs of treewidth k.

What carries the argument

The c-clustered set, a vertex subset inducing only components of size at most c, selected inductively along a tree decomposition of width k.

If this is right

  • The result improves the bound previously obtained by Dvořák and Wood.
  • It confirms the conjecture of Chappell and Pelsmajer exactly when c is at most 2 or k equals 1.
  • For c equals 3 and k equals 2 the lower bound of 5/9 n is tight.
  • In general the factor c over (c plus k plus one) cannot be replaced by the larger factor c over (c plus k) because of the given constructions.

Where Pith is reading between the lines

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

  • The same inductive selection technique on tree decompositions could be tested on related width measures such as pathwidth or branchwidth.
  • The guaranteed existence of large c-clustered sets may support dynamic-programming algorithms that solve clustering or partitioning tasks exactly on bounded-treewidth input graphs.

Load-bearing premise

The argument assumes a tree decomposition of width k exists and that vertices can be chosen inductively so that induced components in the remaining graph stay small.

What would settle it

Any explicit graph of treewidth k whose largest c-clustered set has size strictly less than c n over (c plus k plus one) would disprove the general lower bound.

read the original abstract

A set $S\subseteq V$ of vertices of a graph $G$ is a $c$-clustered set if it induces a subgraph with components of order at most $c$ each, and $\alpha_c(G)$ denotes the size of a largest $c$-clustered set. For any graph $G$ on $n$ vertices and treewidth $k$, we show that $\alpha_c(G) \geq \frac{c}{c+k+1}n$, which improves a result of Dvo\v{r}{\'a}k and Wood [Innov.\ Graph Theory, 2025], while we construct $n$-vertex graphs $G$ of treewidth $k$ with $\alpha_c(G)\leq \frac{c}{c+k}n$. In the case $c\leq 2$ or $k=1$ we prove the better lower bound $\alpha_c(G) \geq \frac{c}{c+k}n$, which settles a conjecture of Chappell and Pelsmajer [Electron.\ J.\ Comb., 2013] and is best-possible. Finally, in the case $c=3$ and $k=2$, we show $\alpha_c(G) \geq \frac{5}{9}n$ which is best-possible.

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

0 major / 1 minor

Summary. The paper proves that for any graph G on n vertices with treewidth k, the c-clustered independence number satisfies α_c(G) ≥ [c / (c + k + 1)] n. It improves a prior result of Dvořák and Wood, and provides matching constructions showing α_c(G) ≤ [c / (c + k)] n is possible. For the cases c ≤ 2 or k = 1 the lower bound improves to [c / (c + k)] n (settling a conjecture of Chappell and Pelsmajer and shown tight); for c = 3 and k = 2 the bound is 5n/9 and is tight.

Significance. If the proofs hold, the work supplies tight or near-tight bounds on clustered independence numbers in bounded-treewidth graphs, resolving an open conjecture in the special cases and improving the general bound. The constructions establish optimality where claimed, and the inductive selection along tree decompositions is a standard technique that appears to be applied correctly to control component sizes.

minor comments (1)
  1. The abstract cites 'Innov. Graph Theory, 2025' and 'Electron. J. Comb., 2013'; expanding the journal names in the reference list would improve readability.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment of the manuscript, including the summary of our results on c-clustered independence numbers in graphs of treewidth k, and for recommending acceptance.

Circularity Check

0 steps flagged

No significant circularity; derivation is self-contained

full rationale

The paper presents direct inductive proofs over tree decompositions to establish lower bounds on α_c(G), with matching constructions for upper bounds in special cases. No equations or definitions reduce the claimed bounds to fitted parameters, self-referential quantities, or load-bearing self-citations; the results improve external prior work (Dvořák-Wood, Chappell-Pelsmajer) via standard separator arguments without importing uniqueness theorems or ansatzes from the authors' own prior papers. The central claims rest on explicit inductive selection controlling component sizes, which is independent of the target bound itself.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on standard definitions of treewidth via tree decompositions and of c-clustered sets; no free parameters, invented entities, or ad-hoc axioms appear in the abstract.

axioms (2)
  • standard math Treewidth k is witnessed by a tree decomposition of width k
    Invoked implicitly when the paper states the bound for any graph of treewidth k
  • standard math Induced subgraphs on components of size at most c define a c-clustered set
    Direct from the definition given in the abstract

pith-pipeline@v0.9.0 · 5754 in / 1522 out tokens · 49548 ms · 2026-05-24T09:51:26.373714+00:00 · methodology

discussion (0)

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Reference graph

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