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

TL;DR

This paper introduces a new partially ordered structure among factor-components of graphs with perfect matchings and generalizes the canonical partition, providing unique, polynomial-time computable decompositions for such graphs.

Contribution

It reveals a partially ordered structure and extends Kotzig's canonical partition to general graphs with perfect matchings, with proofs of canonicity and polynomial-time algorithms.

Findings

Established a partially ordered structure among factor-components.

Generalized Kotzig's canonical partition for broader graph classes.

Proved polynomial-time computability of the decompositions.

Abstract

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.