Forcing and anti-forcing edges in bipartite graphs

Let G be a connected bipartite graph with a perfect matching and the minimum degree at least two. The concept of an anti-forcing edge in G was introduced by Li in Li (1997). One known generalized version for an anti-forcing edge is an anti-forcing set S, which is a set of edges of G such that the spanning subgraph G−S has a unique perfect matching. In this paper, we introduce a new generalization of an anti-forcing edge: an anti-forcing path and an anti-forcing cycle. We show that the existence of an anti-forcing edge in G is equivalent to the existence of an anti-forcing path or an anti-forcing cycle in G. Then we show that G has an edge that is both forcing and anti-forcing if and only if G is an even cycle. In addition, e-anti-forcing paths and e-anti-forcing cycles in hexagonal systems are identified. The parallel concepts of forcing-paths and forcing-cycles of G are also presented.