M-alternating paths and the construction of defect n-extendable bipartite graphs with different connectivities

A near perfect matching is a matching covering all but one vertex in a graph. Let G be a connected graph and n ≤ ( | V ( G ) | − 2 ) / 2 be a positive integer. If any n independent edges in G are contained in a near perfect matching, then G is said to be defect n -extendable. In this paper, we first characterize defect n -extendable bipartite graph G with n = 1 or κ ( G ) ≥ 2 respectively using M -alternating paths. Furthermore, we present a construction characterization of defect n -extendable bipartite graph G with n ≥ 2 and κ ( G ) = 1 . It is also shown that these characterizations can be transformed to polynomial time algorithms to determine if a given bipartite graph is defect n -extendable.