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We investigate the set of excluded minors of connectivity 2 for the class of frame matroids. We exhibit a list $mathcal{E}$ of 18 such matroids, and show that if $N$ is such an excluded minor, then either $N in mathcal{E}$ or $N$ is a 2-sum of $U_{2,4}$ and a 3-connected non-binary frame matroid.
Frame matroids and lifted-graphic matroids are two distinct minor-closed classes of matroids, each of which generalises the class of graphic matroids. The class of quasi-graphic matroids, recently introduced by Geelen, Gerards, and Whittle, simultane
We show that the class of bicircular matroids has only a finite number of excluded minors. Key tools used in our proof include representations of matroids by biased graphs and the recently introduced class of quasi-graphic matroids. We show that if $
The cut-rank of a set $X$ in a graph $G$ is the rank of the $Xtimes (V(G)-X)$ submatrix of the adjacency matrix over the binary field. A split is a partition of the vertex set into two sets $(X,Y)$ such that the cut-rank of $X$ is less than $2$ and b
Given a 3-connected biased graph $Omega$ with a balancing vertex, and with frame matroid $F(Omega)$ nongraphic and 3-connected, we determine all biased graphs $Omega$ with $F(Omega) = F(Omega)$. As a consequence, we show that if $M$ is a 4-connected
Let $M$ be a 3-connected matroid and let $mathbb F$ be a field. Let $A$ be a matrix over $mathbb F$ representing $M$ and let $(G,mathcal B)$ be a biased graph representing $M$. We characterize the relationship between $A$ and $(G,mathcal B)$, settlin