Improving a chain theorem for triangle-free 3-connected matroids

In 2007, Kriesell established a chain theorem for triangle-free 3-connected graphs. Any triangle-free 3-connected graph can be reduced to a double-wheel or to K3,3 by performing a sequence of simple operations without leaving the class of triangle-free 3-connected graphs. Double-wheels define the only infinite family of graphs that are irreducible with respect to these simple operations. In 2013, Lemos extended Kriesell's theorem for matroids. In this case, there are four infinite families of irreducible matroids. In this paper, we improve these results by proving that one of Kriesell's reduction operations can be avoided provided the number of families of irreducible matroids is increased by four.