Minimally non-Pfaffian graphs

We consider the question of characterizing Pfaffian graphs. We exhibit an infinite family of non-Pfaffian graphs minimal with respect to the matching minor relation. This is in sharp contrast with the bipartite case, as Little [C.H.C. Little, A characterization of convertible ( 0 , 1 ) -matrices, J. Combin. Theory Ser. B 18 (1975) 187–208] proved that every bipartite non-Pfaffian graph contains a matching minor isomorphic to K 3 , 3 . We relax the notion of a matching minor and conjecture that there are only finitely many (perhaps as few as two) non-Pfaffian graphs minimal with respect to this notion. ine Pfaffian factor-critical graphs and study them in the second part of the paper. They seem to be of interest as the number of near perfect matchings in a Pfaffian factor-critical graph can be computed in polynomial time. We give a polynomial time recognition algorithm for this class of graphs and characterize non-Pfaffian factor-critical graphs in terms of forbidden central subgraphs.