Z-transformation graphs of perfect matchings of plane bipartite graphs

Abstract

Let G be a plane bipartite graph with at least two perfect matchings. The Z-transformation graph, Z F ( G), of G with respect to a specific set F of faces is defined as a graph on the perfect matchings of G such that two perfect matchings M 1 and M 2 are adjacent provided M 1 and M 2 differ only in a cycle that is the boundary of a face in F. If F is the set of all interior faces, Z F ( G) is the usual Z-transformation graph; If F contains all faces of G it is a novel graph and called the total Z-transformation graph. In this paper, we give some simple characterizations for the Z-transformation graphs to be connected by applying the above new idea. Furthermore, we show that the total Z-transformation graph of G is 2-connected if G is elementary; the total Z-transformation digraph of G is strongly connected if and only if G is elementary.