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arXiv · 2604.04896

Measuring Depth of Matroids

Jakub Balabán·Petr Hliněný·Jan Jedelský·Kristýna Pekárková
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Abstract

Motivated by recently discovered connections between matroid depth measures and block-structured integer programming [ICALP 2020, 2022], we undertake a systematic study of recursive depth parameters for matrices and matroids, aiming to unify recently introduced and scattered concepts. We propose a general framework that naturally yields eight different depth measures for matroids, prove their fundamental properties and relationships, and relate them to two established notions in the field: matroid branch-depth and a newly introduced natural depth counterpart of matroid tree-width. In particular, we show that six of our eight measures are mutually functionally inequivalent, and among these, one is functionally equivalent to matroid branch-depth and another to matroid tree-depth. Importantly, we also prove that these depth measures coincide on matroids and on matrices over any field, which is (somehow surprisingly) not a trivial task. Finally, we provide a comparison between the matroid parameters and classical depth measures of graphs.

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Jakub Balabán, Petr Hliněný, Jan Jedelský, Kristýna Pekárková. 2026-04-06. Measuring Depth of Matroids. https://arxiv.org/abs/2604.04896

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