  Given a graph, a barrier is a set of vertices determined by the Berge
formula---the min-max theorem characterizing the size of maximum matchings. The
notion of barriers plays important roles in numerous contexts of matching
theory, since barriers essentially coincides with dual optimal solutions of the
maximum matching problem. In a special class of graphs called the elementary
graphs, the family of maximal barriers forms a partition of the vertices; this
partition was found by Lov\'asz and is called the canonical partition. The
canonical partition has produced many fundamental results in matching theory,
such as the two ear theorem. However, in non-elementary graphs, the family of
maximal barriers never forms a partition, and there has not been the canonical
partition for general graphs. In this paper, using our previous work, we give a
canonical description of structures of the odd-maximal barriers---a class of
barriers including the maximal barriers---for general graphs; we also reveal
structures of odd components associated with odd-maximal barriers. This result
of us can be regarded as a generalization of Lov\'asz's canonical partition.
