Connectivity gaps among matroids with the same enumerative invariants

• DOI: 10.1016/j.aam.2023.102648
• Publication Date: 2023-12-08T16:20:37Z
• Authors: Joseph E. Bonin; Kevin Long
• Keywords: FOS: Mathematics; Mathematics - Combinatorics; Combinatorics (math.CO); 0101 mathematics; 05B35; 01 natural sciences

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.