A Note on the Connectivity of Binary Matroids

Abstract

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.