Dulmage–Mendelsohn decomposition

In graph theory, the Dulmage–Mendelsohn decomposition is a partition of the vertices of a bipartite graph into subsets, with the property that two adjacent vertices belong to the same subset if and only if they are paired with each other in a perfect matching of the graph. It is named after A. L. Dulmage and Nathan Mendelsohn, who published it in 1958.

Source: Wikipedia — Dulmage–Mendelsohn decomposition (CC BY-SA 4.0)