A Note on the Connectivity of Binary Matroids - Yin, Jun; Huo, Bofeng; Lai, Hong-Jian (Discontinuity, Nonlinearity, and Complexity, 2026)
In [J. Combinatorial Theory, Ser. B, 28 (1980), 305-359], Seymour introduced the binary matroid 3-sums and proved that if a 3-connected binary matroid M is a 3-sum of matroids M_1 and M_2, then each of M_1 and M_2 is isomorphic to a proper minor of M. For a 3-connected binary matroid M expressed as a 3-sum of M_1 and M_2, we show that in general, both M_1 and M_2 are 2-connected, and if M_1 and M_2 are simple matroids, then both M_1 and M_2 are also 3-connected.