Complex sentence: A matching that covers all but one vertex of a graph is called a near-perfect matching, so a factor-critical graph is a graph in which there are near-perfect matchings that avoid every possible vertex.
Split into simpler sentences: A matching that covers all but one vertex of a graph is called a near-perfect matching. So equivalently, a factor-critical graph is a graph in which there are near-perfect matchings that avoid every possible vertex.