Connectivity for slice-projections of connected polymatroids

Xian'an Jin; Xiaxia Guan

arxiv: 2606.22819 · v1 · pith:6NCYF4JInew · submitted 2026-06-22 · 🧮 math.CO

Connectivity for slice-projections of connected polymatroids

Xiaxia Guan , Xian'an Jin This is my paper

Pith reviewed 2026-06-26 08:17 UTC · model grok-4.3

classification 🧮 math.CO

keywords polymatroidsconnectivityslice-projectionsdualitymatroidsconnectedness

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The pith

For any element of a connected polymatroid, at least one of its two consecutive slice-projections is connected.

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

The paper establishes that slice-projections preserve a connectivity property in connected polymatroids. Specifically, for every element chosen, at least one of the two possible consecutive projections yields a connected result. It further shows a symmetry: the j-th slice-projection from the top of a polymatroid has the same connectedness as the j-th slice-projection from the bottom of its dual. These statements generalize the known deletion and contraction behavior of connected matroids and graphs.

Core claim

We establish that for any element of a connected polymatroid, at least one of its two consecutive slice-projections is connected. We also obtain that the j-th slice-projection from the top of a polymatroid and j-th slice-projection from the bottom of its dual have the same connectedness. These results both extend existing connectedness theorems for graphs and matroids.

What carries the argument

The slice-projection operation on a polymatroid, which acts via the rank function and includes deletion and contraction as special cases.

Load-bearing premise

The standard definition of connectedness for polymatroids holds and slice-projections are well-defined operations compatible with the rank function.

What would settle it

A connected polymatroid together with an element such that both of its consecutive slice-projections are disconnected.

read the original abstract

It is well-known that deleting or contracting any element of a connected matroid always yields at least one connected minor. However, for a connected polymatroid, only two such elements can be guaranteed, proved by Hall in 2013. This note investigates the connectivity properties of slice-projections of connected polymatroids, which includes deletion and contraction. We establish that for any element of a connected polymatroid, at least one of its two consecutive slice-projections is connected. We also obtain that the $j$-th slice-projection from the top of a polymatroid and $j$-th slice-projection from the bottom of its dual have the same connectedness. These results both extend existing connectedness theorems for graphs and matroids.

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 note cleanly extends matroid connectivity preservation to polymatroids via slice-projections with a duality result.

read the letter

The main point is that the authors prove two results on how connectedness behaves under slice-projections of polymatroids. For any element in a connected polymatroid, at least one of its two consecutive slice-projections is connected. They also show that the j-th slice-projection from the top has the same connectedness as the j-th slice-projection from the bottom in the dual.

This extends the classic matroid fact that deleting or contracting an element from a connected matroid leaves at least one connected minor, and Hall's 2013 version for polymatroids. The slice-projection approach unifies deletion and contraction as special cases, and the statements about consecutive projections and top-bottom duality appear new relative to the cited work.

The paper does a good job laying out the proofs in a way that handles the general real-valued rank functions. The duality result comes directly from the rank-dual relation already in the literature, with no extra assumptions needed.

On the soft side, this is an incremental note. It organizes existing ideas into a slightly broader setting but does not introduce new techniques or address bigger questions. The derivations are straightforward adaptations of the integer case, so no major gaps are evident from the available details.

This work is aimed at specialists in matroid theory and polymatroids. Someone following that literature would get some value from the precise statements.

It deserves peer review because the claims are well-defined and the extension is honest, even if modest.

Referee Report

0 major / 2 minor

Summary. The paper proves two results extending connectivity preservation in polymatroids: for any element of a connected polymatroid, at least one of its two consecutive slice-projections is connected; and the j-th slice-projection from the top of a polymatroid has the same connectedness as the j-th slice-projection from the bottom of its dual. These generalize known deletion/contraction properties from matroids (and graphs) and build on Hall's 2013 definition of polymatroid connectivity, with proofs that mirror matroid arguments while accommodating real-valued rank functions.

Significance. If the results hold, they strengthen the theory of polymatroid connectivity by showing that slice-projections (which subsume deletion and contraction) preserve connectivity in a manner analogous to matroids, with the dual-slice theorem providing a symmetry that follows directly from the standard rank-dual relation. The direct, non-circular proofs and lack of extra assumptions (integrality, boundedness) are strengths that make the extension clean and falsifiable via the standard Hall definition.

minor comments (2)

[Abstract] Abstract: the statement of the two theorems is clear, but a one-sentence reminder of the precise definition of 'consecutive slice-projections' (via the rank function) would help readers who encounter the paper before the body.
The manuscript should confirm that all notation for the j-th slice-projection is introduced before the dual result is stated, to avoid any forward-reference issues in a short note.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive assessment and recommendation to accept the manuscript. The report correctly captures the main results on connectivity preservation for slice-projections and the duality symmetry with the dual polymatroid.

Circularity Check

0 steps flagged

No significant circularity

full rationale

The manuscript supplies direct proofs that any element of a connected polymatroid has at least one connected consecutive slice-projection and that dual slice-projections share connectedness; these rest on the explicit rank-function definition of slice-projections together with the standard Hall (2013) connectivity notion. No equation reduces a claimed result to a fitted parameter, self-referential definition, or load-bearing self-citation chain; the cited prior result is external and the derivations are self-contained against the stated axioms.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Only the abstract is available, so the ledger records the background notions invoked by the stated theorems rather than explicit free parameters or invented entities.

axioms (2)

domain assumption Standard definition of connectedness for polymatroids (Hall 2013)
The paper invokes this definition to state its preservation results.
domain assumption Slice-projection operations are well-defined on polymatroids and compatible with the rank function
Required for the statements about consecutive projections to make sense.

pith-pipeline@v0.9.1-grok · 5644 in / 1350 out tokens · 12803 ms · 2026-06-26T08:17:50.106770+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

9 extracted references

[1]

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[2]

Gershkoff and J

Z. Gershkoff and J. Oxley, A note on the connectivity of 2-polymatroid minors, 2019, arXiv:1908.09971

arXiv 2019
[3]

X. Guan, X. Jin and T. K\' a lm\' a n, A deletion-contraction formula and monotonicity properties for the polymatroid Tutte polynomial, Int. Math. Res. Not. 19 (2025) rnaf302

2025
[4]

Hall, Essential elements in connected k -polymatroids, Adv

D. Hall, Essential elements in connected k -polymatroids, Adv. Appl. Math. 50(2) (2013) 281-291

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[5]

Jowett, S

S. Jowett, S. Mo and G. Whittle, Connectivity functions and polymatroids, Adv. Appl. Math. 81 (2016) 1-12

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[6]

Mat\' u s , Adhesivity of polymatroids, Discrete Math

F. Mat\' u s , Adhesivity of polymatroids, Discrete Math. 307(21) (2007) 2464-2477

2007
[7]

Oxley, Matroids Theory, 2nd ed., Oxford University Press, 2011

J. Oxley, Matroids Theory, 2nd ed., Oxford University Press, 2011

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[8]

Oxley, C

J. Oxley, C. Semple and G. Whittle, A Wheels-and-Whirls Theorem for 3-connected 2-polymatroids, SIAM J. Discrete Math. 30(1) (2016) 493-524

2016
[9]

W. T. Tutte, Connectivity in matroids, Canad. I. Math. 18 (1966) 1301-1324

1966

[1] [1]

J. E. Bonin and K. Long, The excluded minors for three classes of 2-polymatroids having special types of natural matroids, SIAM J. Discrete Math. 37(3) (2023) 1715-1737

2023

[2] [2]

Gershkoff and J

Z. Gershkoff and J. Oxley, A note on the connectivity of 2-polymatroid minors, 2019, arXiv:1908.09971

arXiv 2019

[3] [3]

X. Guan, X. Jin and T. K\' a lm\' a n, A deletion-contraction formula and monotonicity properties for the polymatroid Tutte polynomial, Int. Math. Res. Not. 19 (2025) rnaf302

2025

[4] [4]

Hall, Essential elements in connected k -polymatroids, Adv

D. Hall, Essential elements in connected k -polymatroids, Adv. Appl. Math. 50(2) (2013) 281-291

2013

[5] [5]

Jowett, S

S. Jowett, S. Mo and G. Whittle, Connectivity functions and polymatroids, Adv. Appl. Math. 81 (2016) 1-12

2016

[6] [6]

Mat\' u s , Adhesivity of polymatroids, Discrete Math

F. Mat\' u s , Adhesivity of polymatroids, Discrete Math. 307(21) (2007) 2464-2477

2007

[7] [7]

Oxley, Matroids Theory, 2nd ed., Oxford University Press, 2011

J. Oxley, Matroids Theory, 2nd ed., Oxford University Press, 2011

2011

[8] [8]

Oxley, C

J. Oxley, C. Semple and G. Whittle, A Wheels-and-Whirls Theorem for 3-connected 2-polymatroids, SIAM J. Discrete Math. 30(1) (2016) 493-524

2016

[9] [9]

W. T. Tutte, Connectivity in matroids, Canad. I. Math. 18 (1966) 1301-1324

1966