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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, simultaneously generalises both the classes of frame and lifted-graphic matroids. Let $mathcal{M}$ be one of these three classes, and let $r$ be a positive integer. We show that $mathcal{M}$ has only a finite number of excluded minors of rank $r$.
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.
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 $
A class of graphs is $chi$-bounded if there exists a function $f:mathbb Nrightarrow mathbb N$ such that for every graph $G$ in the class and an induced subgraph $H$ of $G$, if $H$ has no clique of size $q+1$, then the chromatic number of $H$ is less
We show that for pairs $(Q,R)$ and $(S,T)$ of disjoint subsets of vertices of a graph $G$, if $G$ is sufficiently large, then there exists a vertex $v$ in $V(G)-(Qcup Rcup Scup T)$ such that there are two ways to reduce $G$ by a vertex-minor operatio
The extremal function $c(H)$ of a graph $H$ is the supremum of densities of graphs not containing $H$ as a minor, where the density of a graph $G$ is the ratio of the number of edges to the number of vertices. Myers and Thomason (2005), Norin, Reed,