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

Vasily Buslov

TL;DR

The paper proves that the restriction of a minimal-weight $k$-component directed spanning forest to any atom of the subset algebra generated by the vertex-sets of trees in $k$-component minimal forests is itself a tree, while noting this property does not hold for forests with fewer components. It develops an arc-replacement framework, constructs subset algebras and their atoms, and analyzes convexity inequalities between minimal forest weights to establish a main theorem via a contradiction-based proof, with careful treatment of labeled versus unlabeled atoms. The work further shows how these results extend across algebras, simplifies in unweighted and undirected cases, and connects to diffusion-type Markov chains under small perturbations by enabling reduced-state representations via atomic trees and minimal forests. The findings provide a structural decomposition that facilitates computation of minimal forests and informs the analysis of limiting dynamics in stochastic processes, with potential applications in combinatorial optimization and coarse-grained Markov models.

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.

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

TL;DR

The paper proves that the restriction of a minimal-weight -component directed spanning forest to any atom of the subset algebra generated by the vertex-sets of trees in -component minimal forests is itself a tree, while noting this property does not hold for forests with fewer components. It develops an arc-replacement framework, constructs subset algebras and their atoms, and analyzes convexity inequalities between minimal forest weights to establish a main theorem via a contradiction-based proof, with careful treatment of labeled versus unlabeled atoms. The work further shows how these results extend across algebras, simplifies in unweighted and undirected cases, and connects to diffusion-type Markov chains under small perturbations by enabling reduced-state representations via atomic trees and minimal forests. The findings provide a structural decomposition that facilitates computation of minimal forests and informs the analysis of limiting dynamics in stochastic processes, with potential applications in combinatorial optimization and coarse-grained Markov models.

Abstract

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

Paper Structure

This paper contains 20 sections, 17 theorems, 29 equations, 10 figures.

Table of Contents

  1. Notations and definitions
  2. Some properties of the arc replacement operation
  3. Subset algebras
  4. Subset algebra generated by a set of forests
  5. Subset algebras generated by minimal forests
  6. Formulation of the main statement
  7. The case of equality in the system of convexity inequalities
  8. Strict inequality in the system of convexity inequalities
  9. Restrictions on arcs outgoing an atom
  10. Connected components of the restriction of a minimal forest to atoms of algebra $\mathfrak{A}_k$
  11. Building a forest with special properties
  12. Simple properties of trees (forests)
  13. Tree partition of a set of tree (forest) vertices
  14. Forest required
  15. Full justification of the hypothesis
  16. ...and 5 more sections

Key Result

Lemma 1

V6 Let $F$ and $G$ be forests, ${\cal V}F={\cal V}G={\cal N}$, ${\cal D}\subset{\cal N}$. Then graph $F_{\uparrow\cal D}^G$ is a forest if in $F$ the set ${\cal N}^{in}_{\cal D}(F)$ is unreachable from ${\cal N}^{out}_{\cal D}(G)$.

Figures (10)

  • Figure 1: Possible arrangement of 3 atoms in trees of a forest, if they are not all located in one tree at once.
  • Figure 2: Subgraph $F|_{\cal U}$ induced by an atom ${\cal U}\in\mathfrak{A}_k$. ${\cal X}$ is a set of vertices of a connected component including the root $x$, ${\cal Y}={\cal U}\setminus {\cal X}$. On the left -- the hypothesis is unfair. On the right -- the hypothesis is true, $x$ is the only root of the graph $F|_{\cal U}$, ${\cal X}={\cal U}$.
  • Figure 3: By assumption, in $F$ two arcs outgo from the atom ${\cal U}={\cal X}\cup{\cal Y}$ (these arcs are shown in the figure) and there is a forest in which both ends of these arcs are not in the same tree with ${\cal U}$.
  • Figure 4: If we assume that in some forest the atom ${\cal U}$ has arcs ending in ${\cal E}$ and in ${\cal M}$, then by Lemma 3 there are not forests of the types $a)$ and $b)$, and then by Lemma 2 there must exist forests of both type $c)$ and type $d)$.
  • Figure 5: By assumption, two arcs in $F$ outgo from atom ${\cal U}={\cal X}\cup{\cal Y}$ and one of them enters the labeled atom ${\cal M}$. Then in the forest $Q=G^F_{\uparrow{\cal U}}$ (on the right) the sets ${\cal X}$ and ${\cal Y}$ are in different trees.
  • ...and 5 more figures

Theorems & Definitions (48)

  • Lemma 1
  • Remark 1
  • Lemma 2
  • proof
  • Proposition 1
  • proof
  • Remark 2
  • Theorem 1
  • proof
  • Lemma 3
  • ...and 38 more