On a constructive characterization of a class of trees related to pairs of disjoint matchings

TL;DR

This paper provides a constructive characterization of trees where the maximum size of a matching in pairs of disjoint matchings equals the maximum matching size of the tree, using a new decomposition algorithm.

Contribution

It introduces a novel decomposition algorithm for trees and characterizes trees satisfying the equality between the largest matching in disjoint pairs and the maximum matching.

Findings

Characterization of trees with alpha equals beta

Development of a new tree decomposition algorithm

Proof of the main theorem based on the decomposition

Abstract

For a graph consider the pairs of disjoint matchings which union contains as many edges as possible, and define a parameter $\alpha$ which eqauls the cardinality of the largest matching in those pairs. Also, define $\betta$ to be the cardinality of a maximum matching of the graph. We give a constructive characterization of trees which satisfy the $\alpha$=$\betta$ equality. The proof of our main theorem is based on a new decomposition algorithm obtained for trees.