  We present some Pfaffian identities, which are completely different from the
Pl\"ucker relations. As consequences we obtain a quadratic identity for the
number of perfect matchings of plane graphs, which has a simpler form than the
formula by Yan et al (Graphical condensation of plane graphs: a combinatorial
approach, Theoret. Comput. Sci., to appear), and we also obtain some new
determinant identities.
