On Defect N-Extendable Bipartite Graph and Its Application

A near perfect matching is a matching covering all but one vertex in a graph. Let G be a connected graph with 2n + 3 vertices at least. If any n independent edges in G are contained in a near perfect matching, then G is said to be defect n-extendable. Let G be a defect n-extendable bipartite graph with connectivity not less than two and M be a near perfect matching of G. In this paper, we first characterize G using M-alternating paths, which can be developed into an effective algorithm to determine whether a bipartite graph with connectivity not less than two is defect n-extendable. This problem finds application in the circuit design of allocating jobs to processors when some jobs require specified machines to process. We also study the number of disjoint M-alternating paths between two subsets and between a vertex and a subset of G respectively.