A Partially Ordered Structure and a Generalization of the Canonical Partition for General Graphs with Perfect Matchings

This paper is concerned with structures of general graphs with perfect matchings. We first reveal a partially ordered structure among factor-components of general graphs with perfect matchings. Our second result is a generalization of Kotzig’s canonical partition to a decomposition of general graphs with perfect matchings. It contains a short proof for the theorem of the canonical partition. These results give decompositions which are canonical, that is, unique to given graphs. We also show that there are correlations between these two and that these can be computed in polynomial time.