Factor criticality and complete closure of graphs Original Research Article

A graph G is said to be n-factor-critical if G−T has a perfect matching for every T⊂V(G) with |T|=n. For a vertex x of a graph G, local completion of G at x is the operation of joining every pair of nonadjacent vertices in NG(x). For a property P of graphs, a vertex x in a graph G is said to be P-eligible if the subgraph of G induced by NG(x) satisfies P but it is not complete. For a graph G, a graph H is said to be a P-closure of G if there exists a series of graphs G=G0,G1,…,Gr=H such that Gi is obtained from Gi−1 by local completion at some P-eligible vertex in Gi−1 and H=Gr has no P-eligible vertex. In this paper, we investigate the relation between factor-criticality and a P-closure, where P is a bounded independence number or a bounded domination number.