On excluded minors of connectivity 2 for the class of frame matroids

Abstract

We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list E of 18 such matroids, and show that if N is such an excluded minor, then either N ∈ E or N is a 2-sum of U 2,4 and a 3-connected non-binary frame matroid.

Infinitely many excluded minors for frame matroids and for lifted-graphic matroids

“…Our excluded-minors are based on constructions introduced by Chen and Whittle [3]. DeVos, Funk, and Pivotto [4,5] characterised the non-3-connected excluded-minors for the class of frame matroids.…”

On excluded minors for classes of graphical matroids

“…Bicircular matroids are a relatively well-studied proper minor-closed class of frame matroids; it is known that this class has only a finite number of excluded minors [3]. There are also natural proper minor-closed classes of frame matroids, and of lifted-graphic matroids, that have, for any fixed r ≥ 3, infinitely many excluded minors of rank r [4]. The first systematic study of excluded minors for the class of frame matroids is [5], in which 18 excluded minors of connectivity 2 for the class of frame matroids is exhibited, and it is proved that any other excluded minor of connectivity 2 is a 2-sum of a 3-connected non-binary frame matroid with U 2,4 .…”