  Kuo introduced his 4-point condensation in 2003 for bipartite planar graphs.
In 2006 Kuo generalized this 4-point condensation to planar graphs that are not
necessarily bipartite. His formula expressed the product between the number of
perfect matching of the original graph $G$ and that of the subgraph obtained
from $G$ by removing the four distinguished vertices as a Pfaffian of order 4,
whose entries are numbers of perfect matchings of subgraphs of $G$ obtained by
removing various pairs of vertices chosen from among the four distinguished
ones. The compelling elegance of this formula is inviting of generalization.
Kuo generalized it to $2k$ points under the special assumption that the
subgraph obtained by removing some subset of the $2k$ vertices has precisely
one perfect matching. In this paper we prove that the formula holds in the
general case. We also present a couple of applications.
