On the connectivity function of a binary matroid (Q1850627)

The connectivity function of a matroid \(M\) is defined as \(\xi_M(X,Y)= r_M(X)+ r_M(Y)- r(M)+1\) where \(\{X,Y\}\) is a partition of the ground set \(E(M)\). Cunningham conjectured that if \(M\) and \(N\) are connected matroids on the same ground set having the same connectivity function, then \(N= M\) or \(N= M^*\). \textit{P. D. Seymour} [J. Comb. Theory, Ser. B 45, 25-30 (1988; Zbl 0671.05022)] gave a counterexample to the general case of this conjecture and proved it for the class of binary matroids. In this paper, the author proves that a connected binary matroid is reconstructible, not only in the class of binary matroids, but also in the class of all matroids. That is, if \(M\) is a connected binary matroid and \(N\) is a matroid having the same connectivity function as \(M\) and \(E(M)= E(N)\), then \(N= M\) or \(N= M^*\).