Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Digraphs of potential barriers: properties of their tree structure and algorithm for constructing minimum spanning forests

Vasily Buslov

TL;DR

This work develops a comprehensive framework for analyzing barrier-like structures in weighted graphs by introducing a barrier digraph $V$ and its corresponding potential graph $P$, linked through $v_{ij}=p_{ij}-p_{ii}$. It defines entering forests and trees, introduces key weight notions (tree-like $\lambda$ and forest-like $\mu$ minima), and establishes core identities and transformations (arc replacement, potential shift, and weight replacement) that relate directed and undirected representations. A central contribution is an efficient algorithmic scheme for constructing minimum entering forests of the barrier graph with complexity $O(N^3)$ in dense graphs, including mechanisms to update minima when enlarging components and to handle barriers between tree-sets. The paper also contrasts barrier-graph forests with corresponding potential-graph forests, showing that their minimal structures can differ and motivates a practical road-network analogy to demonstrate the utility of the barrier-potential framework. Overall, the work provides both theoretical foundations and concrete, scalable algorithms for selecting minimum-weight directed forests, with implications for physics-inspired potential models and network design problems.

Abstract

For a weighted digraph without loops $V$, the arc weights of which can be obtained from an undirected graph with loops ${\sf P}$ according to the rule $v_{ij}=p_{ij}-p_{ii}$, the properties are studied. An effective algorithm for constructing directed trees of minimum weight and an algorithm for constructing spanning directed forests of minimum weight are proposed.

Digraphs of potential barriers: properties of their tree structure and algorithm for constructing minimum spanning forests

TL;DR

This work develops a comprehensive framework for analyzing barrier-like structures in weighted graphs by introducing a barrier digraph and its corresponding potential graph , linked through . It defines entering forests and trees, introduces key weight notions (tree-like and forest-like minima), and establishes core identities and transformations (arc replacement, potential shift, and weight replacement) that relate directed and undirected representations. A central contribution is an efficient algorithmic scheme for constructing minimum entering forests of the barrier graph with complexity in dense graphs, including mechanisms to update minima when enlarging components and to handle barriers between tree-sets. The paper also contrasts barrier-graph forests with corresponding potential-graph forests, showing that their minimal structures can differ and motivates a practical road-network analogy to demonstrate the utility of the barrier-potential framework. Overall, the work provides both theoretical foundations and concrete, scalable algorithms for selecting minimum-weight directed forests, with implications for physics-inspired potential models and network design problems.

Abstract

For a weighted digraph without loops , the arc weights of which can be obtained from an undirected graph with loops according to the rule , the properties are studied. An effective algorithm for constructing directed trees of minimum weight and an algorithm for constructing spanning directed forests of minimum weight are proposed.

Paper Structure

This paper contains 26 sections, 6 theorems, 75 equations, 3 figures.

Table of Contents

  1. Definitions and notations
  2. Properties used
  3. Arc replacement operation
  4. Minima of weight on subsets of a vertex set
  5. Tree-like weights on a subset not containing roots
  6. Relatedness of forests
  7. Recalculation of minimum weights when enlarging trees
  8. Digraph of (potential) barriers and graph of potential
  9. Clarification of the use of terminology
  10. Notations and definitions for undirected graphs
  11. Relationship between the barrier graph $V$ and the potential graph P
  12. Potential shift
  13. Removal and giving orientation of trees with replacement of weights
  14. Weights of type $\lambda$ and type $\nu$
  15. On the algorithmization of construction of minimal forests and trees of the barrier graph
  16. ...and 11 more sections

Key Result

Theorem 1

Let $F\in\tilde{\cal F}^k$ and $s\in{\cal K}_F$, then where${\cal S}={\cal V}T^F_s$.

Figures (3)

  • Figure 1: On the left are the arcs of the forest $G$ coming from the vertices of the set ${\cal S}={\cal V}T^F_s$; in the center are the arcs of the tree $T^F_s$ and the set ${\cal D}$ of vertices of the connected component of the induced subgraph $G|_{\cal S}$, including the vertex $s$; on the right are the arcs of the graph $Q=F^G_{\uparrow{\cal D}}$ coming from the vertices of the set ${\cal S}$.
  • Figure 2: Construction of an undirected graph P and an directed graph $V$ based on the potential $P(x)$.
  • Figure 3: Above is the barrier graph $V$ and its spanning minimum entering forest $F$, consisting of two trees; below is the corresponding potential graph P and its spanning minimum forest F, consisting of two trees.

Theorems & Definitions (30)

  • Theorem 1
  • proof
  • Proposition 1
  • proof
  • Theorem 2
  • proof
  • Corollary 1
  • proof
  • Proposition 2
  • proof
  • ...and 20 more