Z-transformation graphs of maximum matchings of plane bipartite graphs

Let G be a plane bipartite graph. The Z-transformation graph Z(G) and its orientation Z̃(G) on the maximum matchings of G are de1ned. If G has a perfect matching, Z(G) and Z̃(G) are the usual Z-transformation graph and digraph. If G has neither isolated vertices nor perfect matching, then Z(G) is not connected. This paper mainly shows that some basic results for Z-transformation graph (digraph) of a plane elementary bipartite graph still hold for every nontrivial component of Z(G) (Z̃(G)). In particular, by obtaining a result that every shortest path of Z(G) from a source of Z̃(G) corresponds to a directed path of Z̃(G), we show that every nontrivial component of Z̃(G) has exactly one source and one sink. Accordingly, it follows that the block graph of every nontrivial component of Z(G) is a path. In addition, we give a simple characterization for a maximum matching of G being a cut-vertex of Z(G). ? 2003 Elsevier B.V. All rights reserved.