Minimally 3-connected binary matroids

A 3-connected matroid MM is said to be minimally 3-connected   if, for any element ee of MM, the matroid M∖eM∖e is not 3-connected. Dawes [R.W. Dawes, Minimally 3-connected graphs, J. Combin. Theory Ser. B 40 (1986) 159–168] showed that all minimally 3-connected graphs can be constructed from K4K4 such that every graph in each intermediate step is also minimally 3-connected. Oxley [J.G. Oxley, On connectivity in matroids and graphs, Trans. Amer. Math. Soc. 265 (1981) 47–58] proved a similar result by giving a characterization of minimally 2-connected matroids. In this paper we generalize Dawes’ result to minimally 3-connected binary matroids. We give a constructive characterization of all minimally 3-connected binary matroids starting from W3W3, the 3-spoked wheel, and F7∗, the Fano dual.