A characterisation of Pfaffian near bipartite graphs

TL;DR

This paper extends Kasteleyn's method for counting perfect matchings to near bipartite graphs, broadening the class of graphs where Pfaffian properties can be characterized.

Contribution

It generalizes Little's characterization of Pfaffian bipartite graphs to include near bipartite graphs, providing a new understanding of their structure.

Findings

Extended Pfaffian characterization to near bipartite graphs

Identified forbidden subgraph conditions for near bipartite graphs

Enhanced enumeration techniques for perfect matchings in complex graphs

Abstract

In 1967 Kasteleyn introduced a powerful method for enumerating the 1-factors of planar graphs. In fact his method can be extended to graphs which permit an orientation under which every alternating circuit is clockwise odd. Graphs with this property are called {\it Pfaffian}. Little characterised Pfaffian bipartite graphs in terms of forbidden subgraphs in 1975. We extend his characterisation to near bipartite graphs.