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arxiv: 2604.11318 · v1 · submitted 2026-04-13 · 🧮 math.CO · cs.DM· cs.DS

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Coarse Balanced Separators in Fat-Minor-Free Graphs

\'Edouard Bonnet, Hung Le, Marcin Pilipczuk, Micha{\l} Pilipczuk

Authors on Pith no claims yet

Pith reviewed 2026-05-10 16:35 UTC · model grok-4.3

classification 🧮 math.CO cs.DMcs.DS
keywords fat minorsbalanced separatorsgraph minorscoarse graph theoryminor-free graphsseparator theoremsgraph algorithmsrandomized algorithms
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The pith

Graphs excluding a fixed d-fat minor admit balanced separators coverable by c n^{1/2+ε} balls of radius r.

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

The paper proves a coarse analogue of the classic theorem that minor-free graphs have O(sqrt(n)) balanced separators. It shows that excluding any fixed graph H as a d-fat minor, where models require branch sets and paths to be mutually distant by at least d, guarantees a balanced separator coverable by c n^{1/2+ε} balls of some fixed radius r. This matters because the separator is small in a metric sense rather than by vertex count, enabling divide-and-conquer methods on broader graph classes. The result holds for weighted graphs and comes with a randomized polynomial-time algorithm that outputs either the separator or a d-fat model of H. A reader cares as it relaxes exact disjointness to distance separation while preserving useful algorithmic consequences.

Core claim

For every integer d, real ε>0, and graph H, constants c and r exist such that every n-vertex graph G excluding H as a d-fat minor has a balanced separator S that can be covered by c n^{1/2+ε} balls of radius r. The claim holds when balance is measured by any vertex weight function, and the proof supplies a randomized polynomial-time algorithm that either returns such an S or produces a d-fat model of H in G.

What carries the argument

d-fat minor: a minor model whose branch sets and connecting paths are pairwise at distance at least d, which serves as the excluded structure forcing the existence of a coarsely coverable balanced separator.

If this is right

  • The separator guarantee holds for arbitrary positive vertex weights rather than uniform counting.
  • A randomized polynomial-time algorithm computes either the desired separator or a forbidden d-fat model.
  • The constants c and r depend only on d, ε, and H, not on n.
  • The result supplies a direct coarse analogue to the O(sqrt(n)) balanced separator theorem for ordinary minor-free graphs.

Where Pith is reading between the lines

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

  • Such coarsely small separators could support divide-and-conquer algorithms for problems like minimum cuts or clustering on fat-minor-free graphs.
  • The approach may extend to other distance-based forbidden structures in metric graph theory.
  • The exponent 1/2+ε might be improvable to 1/2 when r is allowed to depend only on d and H.

Load-bearing premise

The input graph excludes the fixed graph H as a d-fat minor, with all branch sets and paths in any model required to be mutually at distance at least d.

What would settle it

An n-vertex graph with no d-fat model of H whose every balanced separator requires more than n^{1/2+ε} balls of any fixed radius to cover.

read the original abstract

Fat minors are a coarse analogue of graph minors where the subgraphs modeling vertices and edges of the embedded graph are required to be distant from each other, instead of just being disjoint. In this paper, we give a coarse analogue of the classic theorem that an $n$-vertex graph excluding a fixed minor admits a balanced separator of size $O(\sqrt{n})$. Specifically, we prove that for every integer $d$, real $\varepsilon>0$, and graph $H$, there exist constants $c$ and $r$ such that every $n$-vertex graph $G$ excluding $H$ as a $d$-fat minor admits a set $S \subseteq V(G)$ that is a balanced separator of $G$ and can be covered by $c n^{\frac{1}{2}+\varepsilon}$ balls of radius $r$ in $G$. Our proof also works in the weighted setting where the balance of the separator is measured with respect to any weight function on the vertices, and is effective: we obtain a randomized polynomial-time algorithm to compute either such a balanced separator, or a $d$-fat model of $H$ in $G$.

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 / 3 minor

Summary. The paper proves that for every integer d, real ε>0, and graph H, there exist constants c and r such that every n-vertex graph G excluding H as a d-fat minor admits a balanced separator S that can be covered by c n^{1/2+ε} balls of radius r in G. The result extends to the weighted setting (balance measured w.r.t. any vertex weight function) and is effective: a randomized polynomial-time algorithm computes either such an S or a d-fat model of H in G.

Significance. This provides a coarse analogue of the classical balanced-separator theorems (e.g., Lipton-Tarjan, Alon-Seymour-Thomas) for the setting of d-fat minors, where branch sets and paths are required to be pairwise at distance at least d. The win-win argument (either find the forbidden d-fat model or recursively decompose to obtain the separator) is a strength, as is the explicit randomized algorithm that either outputs the separator or certifies the structural obstruction. The dependence of c and r only on d, ε, and H is the expected form for such coarse results and broadens applicability beyond bounded-degree or planar graphs.

minor comments (3)
  1. The abstract and introduction should explicitly note that the constants c and r are permitted to depend on d, ε, and H (as is standard for coarse minor results); this is implicit but worth stating for clarity.
  2. In the description of the randomized algorithm, clarify the success probability and how the sampling is performed inside the recursive decomposition (e.g., which section details the sampling routine).
  3. A brief comparison table or paragraph contrasting the new fat-minor separator bound with the classical O(√n) bound for ordinary minors would help readers situate the result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for the positive assessment, the clear summary of our main theorem, and the recommendation for minor revision. We are pleased that the significance of the coarse balanced-separator result for d-fat-minor-free graphs, the win-win algorithmic approach, and the dependence of the constants only on d, ε, and H were recognized.

Circularity Check

0 steps flagged

No significant circularity; derivation self-contained

full rationale

The paper establishes a new existence result and randomized algorithm for coarse balanced separators in graphs excluding a fixed graph H as a d-fat minor. The argument relies on a win-win decomposition: either a d-fat model of H is found (contradicting the hypothesis) or a recursive procedure yields a separator coverable by c n^{1/2+ε} radius-r balls. This chain does not reduce by the paper's own equations to self-definitions, fitted inputs renamed as predictions, or load-bearing self-citations whose validity depends on the current claim. Constants c and r are permitted to depend on d, ε, and H, which is standard for coarse results. The proof adapts classical separator techniques without circular reduction to inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The result rests on standard graph-theoretic definitions and the existence of constants whose values are not computed explicitly. No free parameters are fitted to data, no new entities are postulated, and no ad-hoc axioms beyond basic graph distance and minor modeling are introduced.

axioms (1)
  • standard math Standard definitions of graphs, vertex/edge distances, and minor models.
    The fat-minor definition and separator notion rely on these background concepts from graph theory.

pith-pipeline@v0.9.0 · 5518 in / 1401 out tokens · 72291 ms · 2026-05-10T16:35:13.233936+00:00 · methodology

discussion (0)

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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. A coarse Menger's Theorem for planar and bounded genus graphs

    math.CO 2026-05 unverdicted novelty 7.0

    In planar and bounded-genus graphs, absence of k pairwise d-far S-T paths implies a vertex set of size f(d,k) whose d-neighborhood intersects every S-T path.

Reference graph

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