Disclosing Barriers: A Generalization of the Canonical Partition Based on Lovász’s Formulation

Abstract

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 Lovász 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 Lovász’s canonical partition.