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 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 any set of fundamental c ircuits can be completed to a cycle basis of Z(M); moreover, for any one-el ement extension M' or. such a matroid M, any cycle basis for Z(M) can be co mpleted to a cycle basis for Z(M'). (C) 1999 Academic Press.