A generalization of Kuo condensation

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 matchings 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 number of applications.