The paper introduces binary functions as a generalization of the cocircuit spaces of binary matroids. It establishes several key results:
The class of stable binary functions is minor-closed, and is characterized by the exclusion of the binary function f^(-1).
The class of rankable binary functions is minor-closed, and is characterized by the exclusion of binary functions f^α where α ≤ -1.
The class of straight binary functions (indicator functions of binary vector spaces) is minor-closed, and is characterized by the exclusion of f^α where α ∈ R{0, 1}, the binary function g on {a, b} where g(X) = 1 for |X| = 0, 1 and g(X) = -1 for |X| = 2, and the binary function f_U2,4 on {a, b, c, d}.
The classes of binary functions corresponding to matroids, binary matroids, and polymatroids are identified and characterized by their excluded minors.
A new proof of Tutte's excluded minor characterization of binary matroids is provided by classifying them within the larger class of binary functions.
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by Benjamin R. ... at arxiv.org 10-03-2024
https://arxiv.org/pdf/2102.01320.pdfDeeper Inquiries