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.
Anti-forcing edge (or, forcing single edge) Forcing edge (or, forcing double edge) Forcing face Plane (minimal)elementary bipartite graph
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