How Trees on Atoms of Subset Algebras Define Minimal Forests and Their Growth

TL;DR

This paper characterizes how minimal trees on atoms of subset algebras define the structure and growth of minimal forests in weighted digraphs, providing bounds and methods for constructing forests with fewer components.

Contribution

It offers a complete description of how minimal trees on atoms of subset algebras influence the form and growth of minimal forests in weighted digraphs, including bounds and construction methods.

Findings

Bounds on extracting tree structure information from minimal trees.

Methods for constructing minimal spanning forests with fewer components.

Insights into how forests grow as the number of arcs increases.

Abstract

A complete description is given of how minimal trees on atoms of the algebra of subsets $\mathfrak{A}_k$ generated by minimal spanning $k$-component forests of a weighted digraph $V$ determine the form of these forests and how forests grow with increasing number of arcs (that is with a decrease in the number of trees). Precise bounds are established on what can be extracted about the tree structure of the original graph if the minimal trees on the atoms of a single algebra $\mathfrak{A}_k$ are known, and also what minimum spanning forests with fewer components can be constructed based on this, and what exactly additional information is required to determine minimum spanning forests consisting of even fewer components.