Base-orderable matroid

Summary

In mathematics, a base-orderable matroid is a matroid that has the following additional property, related to the bases of the matroid.

For any two bases $A$ and $B$ there exists a feasible exchange bijection, defined as a bijection $f$ from $A$ to $B$, such that for every $a\in A\setminus B$, both $(A \setminus \{ a \}) \cup \{f(a)\}$ and $(B \setminus \{ f(a) \}) \cup \{a\}$ are bases.

The property was introduced by Brualdi and Scrimger. A strongly-base-orderable matroid has the following stronger property:

For any two bases $A$ and $B$, there is a strong feasible exchange bijection, defined as a bijection $f$ from $A$ to $B$, such that for every $X\subseteq A$, both $(A \setminus X) \cup f(X)$ and $(B \setminus f(X)) \cup X$ are bases.