A generalization of Kuo condensation

TL;DR

This paper extends Kuo's condensation formula from special cases to a general setting, providing a broader mathematical tool for analyzing perfect matchings in planar graphs with multiple vertices.

Contribution

The paper proves the general case of Kuo's condensation formula for perfect matchings, removing previous restrictions and broadening its applicability.

Findings

Proved the general case of Kuo's condensation formula.

Extended the applicability of the formula to more complex graphs.

Presented applications demonstrating the formula's utility.

Abstract

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.