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We establish the theory of the Dulmage-Mendelsohn decomposition for$b$-matchings. The original Dulmage-Mendelsohn decomposition is a classicalcanonical decomposition of bipartite graphs, which describes the structures ofthe maximum $1$-matchings and the dual optimizers, i.e., the minimum vertexcovers. In this paper, we develop analogical properties, and thus obtain thestructure of the maximum $b$-matchings and characterizes the family of$b$-verifying set.

The Dulmage-Mendelsohn Decomposition for $b$-Matchings