On the maximal matchings of trees

TL;DR

This paper presents an algorithm to count maximal matchings in trees, establishes a lower bound on their number based on the number of vertices, and characterizes trees that achieve this bound.

Contribution

It introduces a new algorithm for counting maximal matchings in trees and characterizes trees with the minimal number of such matchings.

Findings

The number of maximal matchings in any n-vertex tree is at least ceil(n/2).

An explicit characterization of trees that attain this lower bound.

An algorithm for counting maximal matchings in trees.

Abstract

An independent edge set of graph $G$ is a matching, and is maximal if it is not a proper subset of any other matching of $G$. The number of all the maximal matchings of $G$ is denoted by $\Psi(G)$. In this paper, an algorithm to count $\Psi(T)$ for a tree $T$ is given. We show that for any tree $T$ with $n$ vertices, $\Psi(T)\geq\lceil\frac{n}{2}\rceil$, and the tree which obtained the lower bound is characterized.