Cycle Bases for Lattices of Binary Matroids with No Fano Dual Minor and Their One-Element Extensions

In this paper we study the question of existence of a cycle basis (that is, a basis consisting only of cycles) for the lattice Z(M) generated by the cycles of a binary matroid M. We show that, if M has no Fano dual minor, then the set of fundamental circuits of any maximal independent set of M can be completed to a cycle basis of Z(M); moreover, for any one-element extension M 0 of such matroid M, any cycle basis for Z(M) can be completed to a cycle basis for Z(M 0).