Plane Elementary Bipartite Graphs with Forcing or Anti-forcing Edges

Let G be a plane elementary bipartite graph with more than one finite face. We first characterize forcing edges and anti-forcing edges of G in terms of handles and forcing faces. We then introduce the concept of a nice pair of faces of G, which allows us to provide efficient algorithms to construct plane minimal elementary bipartite graphs from G. We show that if G^′ is a minimal elementary spanning subgraph of G, then all vertices of a forcing face of G are contained in a forcing face of G^′, and any forcing face of G is a forcing face of G^′ if it is still a face of G^′. Furthermore, any forcing edge (resp., anti-forcing edge) of G is still a forcing edge (resp., an anti-forcing edge) of G^′ if it is still an edge of G^′. We conclude the paper with the co-existence property of forcing edges and anti-forcing edges for those plane minimal elementary bipartite graphs that remain 2-connected when any handle of length one (if exists) is deleted.