For what matroids $M$ is the intersection graph of the set of all circuits in $M$ connected?


---

Suppose we call matroids with this property "overlapping" then if two matroids $M_1$ and $M_2$ are overlapping and some circuit in $M_1$ is not disjoint to some circuit in $M_2$ then $M_1\oplus M_2$ is also overlapping, thus as a corollary every overlapping matroid must be a connected matroid.