Delta-matroid

A family of subsets of a finite set S is called a delta-matroid if the following symmetric exchange axiom is satisfied:

• If X and Y is in the family, and $$s \in X \Delta Y$$ , then there is an element $$t \in X \Delta Y$$ such that $$X \Delta \{s,t\}$$ is in the family.

Some examples:

• The bases of a matroid form a delta-matroid.
• For any graph, the node sets of matchings form a delta-matroid (this is an example of an even delta-matroid: every member of the family has the same cardinality modulo 2).
• If M is a skew-symmetric matrix, then the row subsets that give non-singular principle submatrices form a delta-matroid (linear delta-matroid).

For more information, see .