A distributive lattice on the set of perfect matchings of a plane bipartite graph

Let G be a plane bipartite graph and M(G) the set of perfect matchings of G. The Z-transformation graph of G is defined as a graph on M(G): M, M' is an element of M(G) are joined by an edge if and only if they differ only in one cycle that is the boundary of an inner face of G. A property that a certain orientation of the Z-transformation graph of G is acyclic implies a partially ordered relation on M(G). An equivalent definition of the poset M(G) is discussed in detail. If G is elementary, the following main results are obtained in this article: the poset M(G) is a finite distributive lattice, and its Hasse diagram is isomorphic to the Z-transformation digraph of G. Further, a distributive lattice structure is established on the set of perfect matchings of any plane bipartite graph.