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arXiv · 2606.22819

Connectivity for slice-projections of connected polymatroids

Xiaxia Guan·Xian'an Jin
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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.

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Xiaxia Guan, Xian'an Jin. 2026-06-22. Connectivity for slice-projections of connected polymatroids. https://arxiv.org/abs/2606.22819

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