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