The binary matroids with no 4-wheel minor

Oxley, James G.

doi:10.1090/S0002-9947-1987-0879563-6

The binary matroids with no $4$-wheel minor
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by James G. Oxley

Trans. Amer. Math. Soc. 301 (1987), 63-75

DOI: https://doi.org/10.1090/S0002-9947-1987-0879563-6

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Abstract:

The cycle matroids of wheels are the fundamental building blocks for the class of binary matroids. Brylawski has shown that a binary matroid has no minor isomorphic to the rank-3 wheel $M({\mathcal {W}_3})$ if and only if it is a series-parallel network. In this paper we characterize the binary matroids with no minor isomorphic to $M({\mathcal {W}_4})$. This characterization is used to solve the critical problem for this class of matroids and to extend results of Kung and Walton and Welsh for related classes of binary matroids.

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