  A near-factor of a finite simple graph $G$ is a matching that saturates all
vertices except one. A graph $G$ is said to be near-factor-critical if the
deletion of any vertex from $G$ results in a subgraph that has a near-factor.
We prove that a connected graph $G$ is near-factor-critical if and only if it
has a perfect matching. We also characterize disconnected near-factor-critical
graphs.
