MathDBFinite excluded-minor conjecture for quasi-graphic matroids
Finite excluded-minor conjecture for quasi-graphic matroids
Let a quasi-graphic matroid be a matroid that has a framework: a graph satisfying conditions (QG1)--(QG4) with respect to the matroid. An excluded minor for the class of quasi-graphic matroids is a matroid not in the class whose every proper minor is quasi-graphic.
Finite excluded-minor conjecture. There are, up to isomorphism, only finitely many excluded minors for the class of quasi-graphic matroids.
The class of quasi-graphic matroids is minor closed and contains both lifted-graphic matroids and frame matroids, each of which generalises the class of graphic matroids. A finite excluded-minor characterisation would describe this class by finitely many forbidden minors; the conjecture remains unresolved here.
Equivalent formulations 1
Other statements of this same problem, merged from separate entries. Each is equivalent to the statement above — proving any one settles them all.
Finite excluded-minor conjecture for quasi-graphic matroids
The class of quasi-graphic matroids contains both the lifted-graphic matroids and the frame matroids. An excluded minor for a minor-closed class is a matroid not in the class whose every proper minor is in the class. Finite excluded-minor conjecture. There are, up to isomorphism, only finitely many excluded-minors for the class of quasi-graphic matroids. The paper's infinite excluded-minor families for the lifted-graphic and frame classes are quasi-graphic, but the authors state that they remain confident that quasi-graphic matroids admit a finite excluded-minor characterization; this conjecture is not resolved here.
source: Rong Chen and Jim Geelen, “Infinitely many excluded minors for frame matroids and for lifted-graphic matroids”, arXiv:1703.04857 (2018).
Sources & referencesView supporting material
Primary source
Rong Chen, “The 9-connected Excluded Minors for the Class of Quasi-graphic Matroids”, arXiv:2106.09955 (2022).
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