On excluded minors for classes of graphical matroids

Abstract

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 M be one of these three classes, and let r be a positive integer. We show that M has only a finite number of excluded minors of rank r.

The $9$-connected Excluded Minors for the Class of Quasi-graphic Matroids

“…Funk and Mayhew [7] recently proved that, for each positive integer r, the class of quasi-graphic matroids has only a finite number of excluded minors of rank r.…”

On graphic elementary lifts of graphic matroids

“…Lifted-graphic matroids play an important role in the matroid minors project of Geelen, Gerards and Whittle [9,10]. Liftedgraphic matroids are studied in [4,5,6,7,19]. This class is minor-closed.…”

“…In [5], it is proved that there exist infinitely many pairwise non-isomorphic excluded minors for the class of lifted-graphic matroids. Frank and Mayhew [7] proved that, for a positive integer r, there are only a finite number of excluded minors of rank r for the class of lifted-graphic matroids. Very less is known about the forbidden-minor characterizations of lifts.…”