Factor-critical graph

In graph theory, a mathematical discipline, a factor-critical graph (or hypomatchable graph) is a graph with an odd number of vertices in which deleting one vertex in every possible way results in a graph with a perfect matching, a way of grouping the remaining vertices into adjacent pairs.

A matching of 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.

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