There are Only a Finite Number of Excluded Minors for the Class of Bicircular Matroids

Matt Devos, Daryl Funk, Luis Goddyn, Gordon Royle

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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 N is an excluded minor of rank at least ten, then N is quasi-graphic. Several small excluded minors are quasi-graphic. Using biased-graphic representations, we find that N already contains one of these. We also provide an upper bound, in terms of rank, on the number of elements in an excluded minor, so the result follows.

Original languageEnglish
JournalAdvances in Combinatorics
Publication statusPublished - 15 Dec 2023

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