The Reconstruction of a Matroid From Its Connectivity Function

In this paper, we shall present an algorithm to decide when a connected matroid M is reconstructible from its connectivity function. When M is not reconstructible, this algorithm gives all the matroids with the same connectivity fucntion as M . 1 Introduction The matroid theory terminology used in this paper will follow Oxley [4]. The connectivity function of a matroid M is dened as M (X; Y ) = r M (X) + r M (Y ) r(M) + 1; where fX; Y g is a partition of E(M ). This function is invariant under duality, since it may be written as M (X; Y ) = r M (X) + r M (X) jX j + 1: In particular, a matroid and its dual have the same connectivity function. When M = M 1 M 2 , then M (X; Y ) + 1 = M1 (X \ E(M 1 ); Y \ E(M 1 )) + M2 (X \ E(M 2 ); Y \ E(M 2 )): The rst author is partially supported by Conselho Nacional de Desenvolvimento Cientco e Tecnologico (CNPq, Brazil). Thus, M 1 M 2 and M 1 M 2 have the same connectivity function. In this paper, we shall adress ..