On two classes of nearly binary matroids

We give an excluded-minor characterization for the class of matroids M in which M ∖ e or M / e is binary for all e in E ( M ) . This class is closely related to the class of matroids in which every member is binary or can be obtained from a binary matroid by relaxing a circuit–hyperplane. We also provide an excluded-minor characterization for the second class.