Irreducible classes and barycentric subdivision on triangle-free 3 connected matroids

Abstract

The 3-connected matroids, fundamental in matroid theory, have two families of irreducible matroids with respect to the operations of deletion and contraction. This result is known as Tutte’s Wheels and Whirls Theorem, established in [11]. Lemos, in [4], considered seven reduction operations to classify the triangles-free 3-connected matroids, five in addition to the two considered by Tutte. The results obtained by Lemos generalize those obtained by Kriesell [2]. Considering only the first three reduction operations defined in [4], we prove that 4 local structures formed by squares and triads behave like "building blocks" for these families of irreducible. Subdividing the seventh reduction, we add another family of triangle-free 3-connected matoids: diamantic matroids. We have established, in a constructive way, that for each matroid in this family there is a unique totally triangular matoid associated. The construction of this one-to-one correspondence is based on the generalized parallel connection and passes through a matroid, unique up to isomorphisms, which corresponds to the barycentric subdivision in the case of graphic matroids.