A note on m-near-factor-critical graphs

Abstract A factor (near-factor) of a finite simple graph G is a matching that saturates all vertices (except one). For m ⩾ 0 , a graph G is said to be m -critical ( m -near-critical) if the deletion of any m vertices from G produces a subgraph that has a factor (near-factor). An m -critical graph is ( m + 1 ) -near-critical. The following results are established. (i) Within the class of ( m + 1 ) -near-critical graphs, a characterization is given for those that are not m -critical. (ii) For an ( m + 1 ) -connected graph, it is ( m + 1 ) -near-critical if and only if it is m -critical. (iii) An ( m + 2 ) -near-critical graph is m -near-critical if its order is at least m + 5 .