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arXiv · 2308.02302

Connectivity gaps among matroids with the same enumerative invariants

Joseph E. Bonin·Kevin Long
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Abstract

Many important enumerative invariants of a matroid can be obtained from its Tutte polynomial, and many more are determined by two stronger invariants, the $\mathcal{G}$-invariant and the configuration of the matroid. We show that the same is not true of the most basic connectivity invariants. Specifically, we show that for any positive integer $n$, there are pairs of matroids that have the same configuration (and so the same $\mathcal{G}$-invariant and the same Tutte polynomial) but the difference between their Tutte connectivities exceeds $n$, and likewise for vertical connectivity and branch-width. The examples that we use to show this, which we construct using an operation that we introduce, are transversal matroids that are also positroids.

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BibTeXRIS

Joseph E. Bonin, Kevin Long. 2023-08-04. Connectivity gaps among matroids with the same enumerative invariants. https://doi.org/10.1016/j.aam.2023.102648

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