Weak Fat Minor Conjecture in Graph Theory

• The Weak Fat Minor Conjecture is a graph-theoretic principle asserting that excluding a sufficiently separated fat minor can force quasi-isometric proximity to a minor-free class.
• Positive results in cases such as K4, K2,t, and tree structures demonstrate robust connections between fat minor exclusion and coarse structural properties.
• Refutations via specific counterexamples reveal general failures of the conjecture, leading to alternative structural insights like coarse separator theorems.

Searching arXiv for the cited papers and related items to ground the article in current arXiv records. The Weak Fat Minor Conjecture is a coarse-graph-theoretic conjectural principle asserting that exclusion of a sufficiently separated copy of a fixed graph should force quasi-isometric proximity to an ordinary minor-excluding class. In its form explicitly described in the literature, it asks whether for every $K\in\mathbb N$ and every graph $H$, there exist $M,A\in\mathbb N$ and a graph $H'$ such that every graph $G$ with no $K$-fat $H$-minor is $(M,A)$-quasi-isometric to a graph with no $H'$-minor (Albrechtsen et al., 21 Aug 2025). This sits between the original Georgakopoulos–Papasoglu fat minor conjecture, which keeps the same target graph $H$, and weaker separator- or width-theoretic consequences of fat-minor exclusion. The conjecture is now known to be false in general (Albrechtsen et al., 21 Aug 2025), but the surrounding theory remains active because several important special cases are positive, several related weakenings survive, and the current frontier is sharply concentrated around a small set of unresolved graphs (Albrechtsen et al., 9 Jan 2026).

1. Definition and formal setting

A $H$0-fat minor of a graph $H$1 in a graph or length space $H$2 is modeled by connected pieces for vertices and edges of $H$3 that must be far apart except at the incidences forced by $H$4. In the graph formulation used in the fat-minor literature, a model $H$5 of $H$6 in $H$7 consists of pairwise disjoint connected branch sets $H$8 for $H$9, and internally disjoint branch paths $M,A\in\mathbb N$0 for $M,A\in\mathbb N$1, where $M,A\in\mathbb N$2 joins the relevant branch sets and avoids all nonincident branch sets. The model is $M,A\in\mathbb N$3-fat if

$M,A\in\mathbb N$4

for every two distinct members $M,A\in\mathbb N$5, except when one is a branch path and the other is an incident endpoint branch set (Albrechtsen et al., 9 Jan 2026). In the length-space formulation, the same principle is expressed via path-connected vertex-pieces $M,A\in\mathbb N$6 and edge-pieces $M,A\in\mathbb N$7, again requiring $M,A\in\mathbb N$8-separation for all nonincident pairs (Davies et al., 2024).

The original conjecture of Georgakopoulos and Papasoglu asked, for every finite graph $M,A\in\mathbb N$9 and every $H'$0, whether there exist $H'$1 such that every graph with no $H'$2-fat $H'$3-minor is $H'$4-quasi-isometric to a graph with no $H'$5-minor (Albrechtsen et al., 9 Jan 2026). The weak variant described later replaces the target obstruction $H'$6 by some possibly larger graph $H'$7 (Albrechtsen et al., 21 Aug 2025). Thus the weak conjecture preserves the philosophy that coarse exclusion of a fat minor should imply coarse proximity to an ordinary minor-closed class, while relaxing the identity of the forbidden minor in the target.

A graph $H'$8 is an asymptotic minor of a space $H'$9 if $G$0 contains a $G$1-fat $G$2-minor for every $G$3, and $G$4 is asymptotically minor-excluded if some finite $G$5 fails to be an asymptotic minor (MacManus, 2024). This vocabulary is closely related but not identical to the weak fat minor conjecture; it is the language in which the group-theoretic and coarse-structural literature often formulates the same large-scale phenomenon.

2. Historical development and conjectural variants

The fat-minor program was motivated by the idea that ordinary minor theory should have a coarse analogue, with quasi-isometry replacing exact graph isomorphism and $G$6-fat minors replacing ordinary minors. Early positive results for specific graphs supported this viewpoint. In particular, there are affirmative cases for $G$7, more generally cycles, $G$8, $G$9, $K$0, $K$1, and, subsequently, all $K$2 (Albrechtsen et al., 2024, Albrechtsen et al., 16 Oct 2025). These results made it plausible that at least some weak form of the general program might survive after failures of the full conjecture.

A separate but related formulation appears in the asymptotic-minor language of Georgakopoulos–Papasoglu. There the conjectural statement recalled in the finitely presented group setting is that a connected, locally finite, quasi-transitive graph $K$3 is asymptotically minor-excluded if and only if it is quasi-isometric to a planar graph (MacManus, 2024). This is not identical to the weak fat minor conjecture as formulated in terms of an arbitrary $K$4, but it belongs to the same conceptual family: exclusion of sufficiently fat finite obstructions should characterize a large-scale structural class.

The literature also distinguishes a much weaker positive statement proved for graphs: if a graph has no $K$5-fat $K$6-minor, then it is quasi-isometric to a graph with no $K$7-fat $K$8-minor (Davies et al., 2024). This is not the weak fat minor conjecture in the sense of quasi-isometry to an ordinary minor-free class, but it is the principal surviving universal thinning theorem after the strong conjectures failed.

This suggests a three-level hierarchy. At the strongest level is the original fat minor conjecture, keeping the same graph $K$9. At an intermediate level is the weak fat minor conjecture, permitting a different ordinary forbidden minor $H$0. At the weakest universal level currently proved is the passage from exclusion of a $H$1-fat $H$2-minor to exclusion of a $H$3-fat $H$4-minor after quasi-isometry (Davies et al., 2024).

3. Refutation in general form

The weak fat minor conjecture is explicitly refuted by the construction of Davies, Hickingbotham, Illingworth, and McCarty. They prove that for every $H$5, there exists a graph $H$6 that does not contain the $H$7-grid as a $H$8-fat minor and is not $H$9-quasi-isometric to a graph with no $(M,A)$0 minor (Albrechtsen et al., 21 Aug 2025). Since the target class “graphs with no $(M,A)$1 minor” is a canonical ordinary minor-closed class, this directly contradicts the weak conjectural principle that forbidding a fat minor should force quasi-isometry to some ordinary minor-free class.

The same paper frames the weak fat minor conjecture as the hope that for every $(M,A)$2 and every graph $(M,A)$3, there exist $(M,A)$4 and a graph $(M,A)$5 such that every graph $(M,A)$6 with no $(M,A)$7-fat $(M,A)$8-minor is $(M,A)$9-quasi-isometric to a graph with no $H'$0-minor (Albrechtsen et al., 21 Aug 2025). Their counterexample uses $H'$1 equal to a fixed planar grid, already at $H'$2. The construction is a slight modification of the Nguyen–Scott–Seymour graphs used to disprove weak coarse Menger-type conjectures, and it simultaneously refutes the conjectured coarse grid theorem of Georgakopoulos and Papasoglu (Albrechtsen et al., 21 Aug 2025).

A related, earlier negative result shows that fat minors cannot generally be thinned all the way down to ordinary minors by quasi-isometries. There exists a finite graph $H'$3 such that for every $H'$4, there is a graph $H'$5 with no $H'$6-fat $H'$7-minor that is not $H'$8-quasi-isometric to any graph with no $H'$9-fat $H$0-minor, and not $H$1-quasi-isometric to any length space with no $H$2-fat $H$3-minor (Davies et al., 2024). That result disproves the original Georgakopoulos–Papasoglu thinning conjecture and establishes that even very weak target exclusions cannot be forced universally.

The negative picture was subsequently sharpened by much smaller counterexamples. The paper "Small counterexamples to the fat minor conjecture" proves incompressibility for $H$4 for all $H$5, for $H$6 with $H$7, and for $H$8 (Albrechtsen et al., 9 Jan 2026). It also shows, for example, that there are graphs with no $H$9-fat $H$00-minor such that every $H$01-quasi-isometric graph has a $H$02-fat $H$03-minor (Albrechtsen et al., 9 Jan 2026). These results directly refute many restricted “weak” variants if those variants include complete graphs $H$04 with $H$05, complete bipartite graphs $H$06 with $H$07, or the planar octahedral graph $H$08.

4. Surviving positive results and exact special cases

Although the weak fat minor conjecture is false in general, a substantial positive theory survives for specific families. One major example is the $H$09 case: every graph with no $H$10-fat $H$11-minor is $H$12-quasi-isometric to a graph with no $H$13-minor (Albrechtsen et al., 2024). The proof proceeds through an honest bounded radial decomposition over a $H$14-minor-free graph, giving explicit parameters such as an honest $H$15-radial decomposition and a resulting quasi-isometry with explicit constants (Albrechtsen et al., 2024). The same framework also yields a new short proof of the corresponding $H$16 case (Albrechtsen et al., 2024).

A second major positive family is $H$17. For every $H$18, the graph $H$19 satisfies the fat minor conjecture of Georgakopoulos and Papasoglu: for every $H$20, there exist $H$21 such that every graph with no $H$22-fat $H$23-minor is $H$24-quasi-isometric to a graph with no $H$25-minor (Albrechtsen et al., 16 Oct 2025). The stronger structural theorem produces an honest bounded graph-partition over a graph $H$26 such that every $H$27-connected multigraph minor of $H$28 lifts to a fat minor upstairs, forcing $H$29 itself to be $H$30-minor-free (Albrechtsen et al., 16 Oct 2025). The paper states an explicit distortion bound of $H$31 (Albrechtsen et al., 16 Oct 2025).

A third positive direction concerns trees. For every finite tree $H$32 and every $H$33, there exist $H$34 such that every graph that does not contain $H$35 as a $H$36-fat minor admits an $H$37-quasi-isometry to a graph with line-width at most $H$38; conversely, for all $H$39 there exist $H$40 and a finite tree $H$41 such that every graph containing $H$42 as a $H$43-fat minor admits no $H$44-quasi-isometry to a graph with line-width at most $H$45 (Nguyen et al., 10 Sep 2025). This does not literally conclude quasi-isometry to an $H$46-minor-free graph, but it gives the tree-case coarse analogue of the Robertson–Seymour path-width characterization, with line-width replacing path-width in the infinite setting.

There is also a strong positive result in geometric group theory. A finitely presented group is asymptotically minor-excluded if and only if some finite index subgroup admits a planar Cayley graph (MacManus, 2024). This is presented as a partial affirmative answer to a conjecture of Georgakopoulos and Papasoglu in the finitely presented group setting. It suggests that the failure of the weak fat minor conjecture in general graphs does not preclude exact characterizations in important quasi-transitive subclasses.

The following table summarizes the status of representative families.

Family or setting	Status	Source
General weak fat minor conjecture	False	(Albrechtsen et al., 21 Aug 2025)
Original thinning to ordinary minors	False in general	(Davies et al., 2024)
$H$47	Positive	(Albrechtsen et al., 2024)
$H$48 for all $H$49	Positive	(Albrechtsen et al., 16 Oct 2025)
Finite trees	Positive width-theoretic analogue	(Nguyen et al., 10 Sep 2025)
Finitely presented groups	Positive planar Cayley characterization	(MacManus, 2024)

These positive cases show that the general failure is not merely a uniform obstruction to all coarse-minor classification. Rather, it indicates that the correct formulation is family-dependent.

5. Structural replacements for the failed conjecture

After the failure of the weak fat minor conjecture, one central direction is to seek structural consequences weaker than quasi-isometry to an ordinary minor-free class. A notable general theorem is a coarse separator result: for every graph $H$50, integer $H$51, and real $H$52, every $H$53-vertex weighted graph that excludes $H$54 as a $H$55-fat minor has a balanced separator that is

$H$56

and there is a randomized polynomial-time algorithm that either finds such a separator or a $H$57-fat model of $H$58 (Bonnet et al., 13 Apr 2026). This gives a coarse analogue of separator theorems for ordinary minor-free graphs, though with an $H$59 loss and with coverability by bounded-radius balls rather than direct cardinality bounds.

The same paper formulates a conjectural sharper separator statement: for every graph $H$60 and $H$61, there should exist constants $H$62 such that every $H$63-vertex weighted graph excluding $H$64 as a $H$65-fat minor has a $H$66-coverable balanced separator (Bonnet et al., 13 Apr 2026). This is best understood as a structural replacement for the false weak fat minor conjecture: rather than quasi-isometric equivalence to a minor-closed class, one asks for coarse separator theory, coarse treewidth bounds, and algorithmic decomposability.

Another universal structural replacement is the power-graph theorem. If a graph $H$67 has no $H$68-fat $H$69-minor, then $H$70 is $H$71-quasi-isometric to the power graph $H$72, and $H$73 has no $H$74-fat $H$75-minor (Davies et al., 2024). This gives a canonical thinning operation that is universally valid, albeit only to the level of $H$76-fat exclusion. The same paper explicitly poses the open problem of whether some $H$77-fat version might still characterize ordinary minor-free classes up to quasi-isometry (Davies et al., 2024).

A plausible implication is that the correct post-counterexample theory is not a single replacement conjecture but a menu of weaker coarse invariants: coarse separators, coarse treewidth, bounded line-width for trees, and family-specific quasi-isometric classifications. The existing theorems support that interpretation, but it remains an inference rather than a formal theorem.

6. Boundary cases, unresolved families, and common misconceptions

A common misconception is that the general disproof renders the positive cases uninformative. The current literature indicates the opposite. The positive $H$78, $H$79, tree, and finitely presented group theorems are exact structural theorems in their own domains (Albrechtsen et al., 2024, Albrechtsen et al., 16 Oct 2025, Nguyen et al., 10 Sep 2025, MacManus, 2024). Their proofs use distinct mechanisms—radial decompositions, bounded graph-partitions, century societies and superfat trees, and group accessibility plus splittings—so they are not simple fragments of a failed general argument.

Another misconception is that all natural weak variants were disproved at once. The paper on small counterexamples makes clear that some cases remain open. Its methods do not refute compressibility of $H$80, $H$81, or, more generally, $H$82, because the Nguyen–Scott–Seymour-based graphs themselves contain $H$83 and $H$84 as arbitrarily fat minors (Albrechtsen et al., 9 Jan 2026). The paper explicitly identifies the Kuratowski graphs

$H$85

as the essential remaining boundary cases for the coarse planarity program (Albrechtsen et al., 9 Jan 2026). Thus any “weak fat minor conjecture” focused on coarse planarity or on $H$86 remains genuinely unresolved.

The relationship with planarity is likewise nuanced. One paper proves that finitely presented groups are asymptotically minor-excluded exactly when some finite index subgroup admits a planar Cayley graph (MacManus, 2024). Another disproves a conjectured coarse grid theorem and the weak fat minor conjecture by excluding a fixed fat planar grid while remaining far from every $H$87-minor-free class (Albrechtsen et al., 21 Aug 2025). These results are not contradictory: the first is a positive theorem in a highly structured quasi-transitive group setting, whereas the second constructs arbitrary graphs with pathological coarse connectivity.

The present frontier may therefore be summarized as follows. The weak fat minor conjecture is false as a universal theorem. Many natural restricted versions are also false, including those covering $H$88 for $H$89, $H$90 for $H$91, and $H$92 (Albrechtsen et al., 9 Jan 2026). Yet several specific families admit full affirmative theorems, and the unresolved strip is concentrated around $H$93, $H$94, and perhaps $H$95 (Albrechtsen et al., 9 Jan 2026).

In that sense, the weak fat minor conjecture now functions less as a live universal conjecture than as a historical organizing idea. Its failure clarified the limits of coarse minor thinning, while the surviving theorems delineate the graph families and ambient settings where a coarse-to-classical transfer principle still holds.