Circuit separation for symmetric matroids

AbstractSymmetric matroids and their associated structure (delta matroids) are a generalization of finite matroids introduced by A. Bouchet to extend several nice properties related to matroids (greedy algorithm, duality, representation). For a given symmetric matroid, the family C of its circuits is said to be weakly separate if the relation “x, y ∈ C for some circuit C” on W is transitive, while C is said to be separate if there exists a transversal V of W such that C ⊆ V or C ∩ V = Ø for any circuit C. These two classes of symmetric matroids are characterized by equivalent properties and excluded minors. Then we give the corresponding interpretation in delta matroids with the characterization of two types of delta matroids coming from matroids, and two results on the symmetric matroids considered as an intersection