Factor-critical graph

In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph) is a graph with n vertices in which every subgraph of n − 1 vertices has a perfect matching. (A perfect matching in a graph is a subset of its edges with the property that each of its vertices is the endpoint of exactly one of the edges in the subset.)

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.