Excluding Kuratowski graphs and their duals from binary matroids

TL;DR

This paper characterizes certain binary matroids excluding specific Kuratowski graphs and their duals, provides algorithms for minor testing, and analyzes growth rates and critical exponents related to these exclusions.

Contribution

It offers a new characterization of internally 4-connected binary matroids without certain minors and introduces a practical algorithm for minor testing.

Findings

Characterization of binary matroids with no M(K3,3)-minor

Algorithm for testing minors in binary matroids

Bound on the critical exponent of such matroids over GF(2)

Abstract

We consider some applications of our characterisation of the internally 4-connected binary matroids with no M(K3,3)-minor. We characterise the internally 4-connected binary matroids with no minor in some subset of {M(K3,3),M*(K3,3),M(K5),M*(K5)} that contains either M(K3,3) or M*(K3,3). We also describe a practical algorithm for testing whether a binary matroid has a minor in the subset. In addition we characterise the growth-rate of binary matroids with no M(K3,3)-minor, and we show that a binary matroid with no M(K3,3)-minor has critical exponent over GF(2) at most equal to four.