On the Connectivity Function of a Binary Matroid

In this paper, we shall consider the following problem: up to duality, is a connected matroid reconstructible from its connectivity function? Cunningham conjectured that this question has an affirmative answer, but Seymour gave a counter-example for it. In the same paper, Seymour proved that a connected binary matroid is always reconstructible from its connectivity function, up to duality, in the class of binary matroids. In this paper, we prove that when M is a connected binary matroid and N another matroid with the same connectivity function as N, then M=N or M=N*.