  A bipartite graph G is known to be Pfaffian if and only if it does not
contain an even subdivision H of $K_{3,3}$ such that $G - VH$ contains a
1-factor. However a general characterisation of Pfaffian graphs in terms of
forbidden subgraphs is currently not known. In this paper we describe a
possible approach to the derivation of such a characterisation. We also extend
the characterisation for bipartite graphs to a slightly more general class of
graphs.
