On Cycle-forced Bipartite Graphs

Let G be a graph with perfect matchings,and let e be an edge of G.If G has only one perfect matching containing e,e is called a forcing edge of G.Let C be an even cycle of G.If G-V(C)has only one perfect,matching.C is called a forcing cycle of G.If every even cycle of G is a forcing cycle,G is called a cycle-forced graph.When G,is a connected bipartite plane graph,let s be a finite face of G If the graph which is obtained from G by deleting all the vertices lying on the boundary of s has only one perfect matching,s is called a forcing face of G.For forcing edges,many results have been obtained,including the characterization of hexagonal system of which each edge is a forcing edge,the characterization of forcing edges in hexagonal system and bipartite plane graph.For forcing faces,there have been some good results,including the characterization of forcing faces in bipartite plane graph,the characterization of bipartite plane graph of which each finite face is a forcing face.For cycle-forced graphs,complete characterizations of cycle-forced hamiltonian bipartite graphs and cycle-forced bipartite graphs which have IS-ears are obtained.In this paper,we study forcing edges,forcing cycles and cycle-forced graphs.The main results are the following:·A complete characterization of cycle-forced bipartite graplhs which have no IS-ears.From this,a complete characterization of cycle-forced bipartite graphs is obtained.·A complete characterization of graphs of which each edge is a forcing edge.·Some characterizations of forcing edges in some special graphs.·A characterization of forcing cycles in bipartite plane graphs.