When a forest, narrowed to an atom of subset algebra, turns out to be a tree

TL;DR

This paper proves that when a directed spanning forest with a specific component structure is restricted to an atom of a certain algebra, it results in a tree, revealing a structural property of minimal forests.

Contribution

It establishes a novel property of minimal directed spanning forests when restricted to atoms of a subset algebra, specifically for forests with exactly k or (k-1) components.

Findings

Restriction of a k-component minimal forest to an algebra atom yields a tree.

This property does not generally hold for forests with fewer components.

The result deepens understanding of the structure of minimal spanning forests.

Abstract

It is proved that the restriction of a $k$ and $(k-1)$-component directed spanning forest of minimal weight to an atom of the subset algebra generated by the sets of vertices of trees of $k$-component minimal spanning forests is a tree. For spanning minimal forests consisting of fewer components, this property, generally speaking, does not exist.