A matroid minor is a matroid obtained by deleting and contracting elements of another matroid. For a finite matroid 
with ground set 
and rank function 
, deletion of 
restricts the independent sets to 
, while contraction of 
is characterized by
 |
for 
. Deleting and contracting
disjoint sets commute, so a minor has the form 
. For a graphic matroid,
these operations correspond to graph edge deletion
and contraction, relating matroid minors to graph minors
(Mizell and Oxley 2026).
A matroid is simple when it has no matroid circuits of size 1 or 2. Mizell and Oxley (2026, Theorem 2) prove that every nonempty finite simple matroid whose matroid cocircuits all have at least four elements has a minor in a specified list of nine minor-minimal simple matroids. Every member of this list has matroid rank at most 5 and at most twelve elements. Under the additional hypothesis of 3-connectivity, eight of them suffice (Mizell and Oxley 2026, Theorem 3).
