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

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.

Significant inaccuracies in the English translation