Verifying (k, 0, d)-extendability in bipartite graphs and its application

A defect d-matching in a graph G is a matching covering all but d vertices in G. Let G = (U, W) be a bipartite graph with bipartition U, W and |W|≥|U|. Let k, d be non-negative integers such that k + d = |W| ≥-|U|. If deleting any k vertices from W, the remaining subgraph of G contains a defect d-matching, then G is said to be (k, 0, d)-extendable. (k, 0, d)-extendable bipartite graphs find applications in designing the robust job assignment circuit. In this paper, we investigate the properties and characterizations of (k, 0, d)-extendable bipartite graphs. Basing on these results, an efficient algorithm to determine the (k, 0, d)-extendability of a bipartite graph is designed and we also prove that the time complexity of the algorithm is much better than that of the algorithm designed basing on the definition.