As a general case of molecular graphs of benzenoid hydrocarbons, we study plane bipartite graphs with Kekule structures (1-factors). A bipartite graph G is called elementary if G is connected and every edge belongs to a 1-factor of G. Some properties of the minimal and the maximal 1-factor of a plane elementary graph are given. A peripheral face f of a plane elementary graph is reducible, if the re- moval of the internal vertices and edges of the path that is the intersection of f and the outer cycle of G results in an elementary graph. We characterize the reducible faces of a plane elementary bipartite graph. This result gen- eralizes the characterization of reducible faces of an elementary benzenoid graph.