The Smallest Classes of Binary and Ternary Matroids Closed under Direct Sums and Complements

Abstract

. The class of cographs or complement-reducible graphs is the class of graphs that can be generated from K 1 using the operations of disjoint union and complementation. By analogy, this paper intro-duces the class of binary comatroids as the class of matroids that can be generated from the empty matroid using the operations of direct sum and taking complements inside of binary projective space. We show that a proper flat of a binary comatroid is a binary comatroid. Our main result identifies those binary non-comatroids for which every proper flat is a binary comatroid. The paper also proves the corresponding results for ternary matroids.