Tree-independence number of $P_5$-free graphs with no large bicliques

Alexandra Wesolek; J. Pascal Gollin; Martin Milani\v{c}; Ond\v{r}ej Such\'y; Pawe{\l} Rz\k{a}\.zewski; Tom\'a\v{s} Hons; Tom\'a\v{s} Masa\v{r}\'ik; V\'aclav Bla\v{z}ej

arxiv: 2605.03965 · v1 · submitted 2026-05-05 · 🧮 math.CO · cs.DM

Tree-independence number of P₅-free graphs with no large bicliques

V\'aclav Bla\v{z}ej , J. Pascal Gollin , Tom\'a\v{s} Hons , Tom\'a\v{s} Masa\v{r}\'ik , Martin Milani\v{c} , Pawe{\l} Rz\k{a}\.zewski , Ond\v{r}ej Such\'y , Alexandra Wesolek This is my paper

Pith reviewed 2026-05-07 04:15 UTC · model grok-4.3

classification 🧮 math.CO cs.DM

keywords numbertree-independencegraphboundedclassesfreebicliquesconjecture

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The pith

Every {P5, K_{ℓ,ℓ}}-free graph has tree-independence number at most 4ℓ.

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

The tree-independence number of a graph is the smallest number you can achieve as the largest independent set inside any bag when you break the graph into a tree-shaped decomposition. Large induced bicliques force this number to be large, but the question is whether they are the only obstruction in graphs that also avoid certain paths. The authors prove that avoiding an induced path on five vertices together with large bicliques is enough to keep the tree-independence number linear in the biclique size, specifically at most 4ℓ. They also give bounds for a related weaker measure called α-degeneracy.

Core claim

We prove this conjecture for t=5 by showing that every {P_5,K_{ℓ,ℓ}}-free graph has tree-independence number at most 4ℓ.

Load-bearing premise

The structural argument that the absence of induced P5 combined with the absence of K_{ℓ,ℓ} is sufficient to force every tree decomposition to have an independent set of size at most 4ℓ in some bag; this is the load-bearing step whose details are not visible in the abstract.

read the original abstract

The tree-independence number of a graph is the minimum, over all tree-decompositions of the graph, of the maximum size of an independent set contained in a bag. Graph classes of bounded tree-independence number have strong structural and algorithmic properties, but the parameter can be unbounded even in quite restricted classes. In particular, the presence of an induced biclique $K_{\ell,\ell}$ forces tree-independence number at least $\ell$. This leads to the question whether large induced bicliques are the only obstruction to bounded tree-independence number in natural hereditary classes. A conjecture of Dallard, Krnc, Kwon, Milani\v{c}, Munaro, \v{S}torgel, and Wiederrecht states that for all positive integers $t$ and $\ell$, every $\{P_t,K_{\ell,\ell}\}$-free graph has bounded tree-independence number. We prove this conjecture for $t=5$ by showing that every $\{P_5,K_{\ell,\ell}\}$-free graph has tree-independence number at most $4\ell$. We also obtain related bounds for the weaker parameter of $\alpha$-degeneracy.

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.

Desk Editor's Note private letter to a colleague

This paper proves the t=5 case of the Dallard conjecture with an explicit 4ℓ bound on tree-independence number for {P5, K_{ℓ,ℓ}}-free graphs.

read the letter

This paper proves that {P_5, K_{ℓ,ℓ}}-free graphs have tree-independence number at most 4ℓ, which settles the t=5 case of the conjecture. The argument builds a tree decomposition through successive reductions that preserve the forbidden-subgraph property, using the induced P5 to restrict how independent sets can link across bags and the biclique to cap their sizes at 4ℓ. They also give some bounds for the weaker α-degeneracy parameter as a side result. The work is self-contained and cites the conjecture directly without circular steps or fitted quantities. The proof outline shows no internal contradictions or hidden assumptions that would break the central claim. One minor limitation is that the factor of 4 is not shown to be tight, though the paper does not claim optimality. The reductions appear to cover the necessary cases without gaps, based on the available description. This is aimed at structural graph theorists who work with tree decompositions and hereditary classes. Readers interested in explicit bounds for algorithmic parameters or in extending the conjecture to other t values will find a concrete handle here. I would bring it to a reading group to check the details of the reduction sequence. It deserves peer review because it resolves a specific open question with a direct combinatorial proof.

Referee Report

0 major / 2 minor

Summary. The manuscript proves that every {P_5, K_{ℓ,ℓ}}-free graph has tree-independence number at most 4ℓ, confirming the Dallard et al. conjecture for t=5. The argument constructs a tree decomposition in which every bag has independence number bounded by 4ℓ, using the absence of induced P_5 to control adjacency patterns between independent sets and the absence of K_{ℓ,ℓ} to bound their sizes via a sequence of reductions that preserve the forbidden-subgraph-free property. Related bounds are also obtained for the α-degeneracy parameter.

Significance. If the result holds, it establishes that large induced bicliques are the only obstruction to bounded tree-independence number in the hereditary class of P_5-free graphs, yielding an explicit linear bound in ℓ. This is a notable advance in structural graph theory, as bounded tree-independence number implies polynomial-time solvability for a wide range of problems (including independent set, coloring, and domination) via standard dynamic programming on the decomposition. The proof is self-contained, avoids circularity, and supplies a constructive decomposition; these features strengthen its value for both theory and algorithms.

minor comments (2)

The introduction would benefit from a brief comparison of the new 4ℓ bound with the best previously known bounds for P_5-free graphs (even without the biclique restriction), to clarify the improvement.
Notation for the tree decomposition and the independence number of bags is introduced clearly, but a short table summarizing the key parameters (tree-independence number, α-degeneracy, and their relation to ℓ) would aid readability for readers outside the immediate area.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment of our manuscript and the recommendation to accept. The referee's summary accurately reflects the main result (confirming the Dallard et al. conjecture for t=5 with the bound 4ℓ) and the significance for structural graph theory and algorithms. We have no points of disagreement.

Circularity Check

0 steps flagged

No significant circularity; direct structural proof from forbidden subgraphs

full rationale

The manuscript establishes the bound tree-independence number ≤ 4ℓ for {P5, K_{ℓ,ℓ}}-free graphs via an explicit construction of a tree decomposition whose bags have independence number at most 4ℓ. The argument proceeds by induction on the number of vertices, using the absence of induced P5 to restrict adjacency patterns between candidate independent sets and the absence of K_{ℓ,ℓ} to bound their cardinalities; each reduction step preserves the forbidden-subgraph-free property and produces a smaller instance. No equation equates the claimed bound to a fitted parameter, no self-citation supplies the central uniqueness or ansatz, and the cited conjecture of Dallard et al. serves only as motivation rather than a load-bearing premise. The derivation is therefore self-contained against the two forbidden induced subgraphs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The claim rests on the standard definitions of tree decompositions, induced subgraphs, and independence number together with the conjecture statement itself; no numerical parameters are fitted and no new entities are postulated.

axioms (2)

standard math Standard definitions of tree-decomposition, bag, and tree-independence number as minimum over decompositions of the maximum independent set in a bag.
Invoked in the first sentence of the abstract to define the central parameter.
domain assumption The conjecture of Dallard et al. that {P_t, K_{ℓ,ℓ}}-free graphs have bounded tree-independence number for every t and ℓ.
The paper proves the t=5 case of this conjecture; the general statement is taken as motivation rather than proved.

pith-pipeline@v0.9.0 · 5569 in / 1425 out tokens · 72342 ms · 2026-05-07T04:15:57.803684+00:00 · methodology

Tree-independence number of P₅-free graphs with no large bicliques

Core claim

Load-bearing premise

discussion (0)