The Binary Matroids Having an Element Which is in Every Four-Wheel Minor

The binary matroids with no three- and four-wheel minors were characterized by Brylawski and Oxley, respectively. The importance of these results is that, in a version of Seymour’s Splitter Theorem, Coullard showed that the three- and four-wheel matroids are the basic building blocks of the class of binary matroids. This paper determines the structure of a class of binary matroids which almost have no four-wheel minor. This class consists of matroids \(M\) having a four-wheel minor and an element \(e\) such that both the deletion and contraction of \(e\) from \(M\) have no four-wheel minor.