Connectivity gaps among matroids with the same enumerative invariants

TL;DR

This paper demonstrates that matroids with identical enumerative invariants like the Tutte polynomial and configuration can still differ significantly in their connectivity properties, revealing gaps in these invariants' descriptive power.

Contribution

It introduces a new operation to construct matroids that share invariants but differ in connectivity, highlighting limitations of these invariants.

Findings

Existence of matroid pairs with same invariants but different connectivity measures

Construction of transversal positroids with these properties

Connectivity differences can be arbitrarily large

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.