Finding the ( n , k , 0)-Extendability in Bipartite Graphs and Its Application

We study the problem of finding the (n,k,0)-extendability in bipartite graphs. Let G be a graph with vertex set V(G). Let n,k,d be non-negative integers such that n + 2k + d ≤ |V(G)| ? 2 and |V(G)| ? n ? d is even. A matching which saturates exactly |V(G)| ? d vertices of G is called a defect-d matching of G. If when deleting any n vertices in V(G) the remaining subgraph contains a matching of k edges and every k-matching can be extended to a defect-d matching, then G is said to be (n,k,d)-extendable. We present an algorithm to find the (n,k,0)-extendability for bipartite graphs. This problem finds application in the circuit design of allocating jobs to processors while some jobs require specified machines to process and some machines are not available. In addition, the connectivity of (n,k,0)-extendable graphs is also discussed.