**Sandra Kingan**, Brooklyn College and the Graduate Center, CUNY

**Authors:**Sandra Kingan

Matroids are a geometry defined by certain combinatorial axioms and like most geometries, some matroids are more important than others. Consider a class of matroids closed under minors. A splitter for the minor-closed class is a 3-connected matroid N in the class, such that no 3-connected matroid in the class has a proper N-minor. A 3-decomposer for the minor-closed class is a 3-connected matroid N in the class with a non-minimal exact 3-separation (A, B), such that any matroid M in the class with an N-minor has a 3-separation (X, Y), where A is a subset of X and B is a subset of Y. Splitters and 3-decomposers are highly technical, but important concepts in matroid structure theory, that capture the structure present in excluded minor classes. I will begin by explaining splitters and decomposers in as simple language as possible. Then I will present a decomposition theorem for binary matroids with no minors isomorphic to S10 or its dual, where S10 is a certain well-known 10-element matroid that plays an important role in the structure of almost-regular matroids. Consequently, a decomposition result for the class of binary matroids with no minors isomorphic to the cycle matroid of the complete bipartite graph on 3 vertices or its dual is easily obtained. These two results imply a 2004 result on internally 4-connected matroids by Zhou (JCTB 91, 327-343) and a 2017 result by Mayhew, Royle, and Whittle (JCTB 125, 95--113).