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How to see the forest for the trees

Erika Bérczi-Kovács, András Frank

TL;DR

The paper surveys the Nash-Williams and Tutte tree-packing and tree-covering theorems and places them in a unifying matroid framework, illustrating how they extend to $k$-partition-connectivity and to the decomposition of graphs, hypergraphs, and directed/mixed structures. It develops constructive characterizations and orientation results, linking partition-connectivity with the existence of multiple disjoint spanning trees, arborescences, and hypertrees, and it discusses algorithmic implications via matroid sums and common-basis problems. The discussion extends to hypergraphs, dypergraphs, and rigidity theory, showing how high connectivity underpins generic rigidity and how recent results sharpen our understanding of partition-deficiency and equitable decompositions. The work highlights interdisciplinary connections, including rigidity, optimization, and game-theoretic switching, underscoring the enduring influence of the tree-packing/covering paradigm in combinatorics and beyond.

Abstract

One of the major starting points of discrete optimization is the theorem of Nash-Williams and Tutte on the existence of $k$ disjoint spanning trees of a graph along with its counterpart on the existence of $k$ forests covering all edges of the graph. These elegant results triggered a comprehensive research that gave rise to far-reaching generalizations and found applications at seemingly far-fetched areas. There are well over a thousand papers in the literature, including quite a few brand-new ones. Our first goal is to enlighten some aspects and links of these developments with the hope that the melody finds its way to non-experts. But we hope that experts will also find some novelties in our orchestration.

How to see the forest for the trees

TL;DR

The paper surveys the Nash-Williams and Tutte tree-packing and tree-covering theorems and places them in a unifying matroid framework, illustrating how they extend to -partition-connectivity and to the decomposition of graphs, hypergraphs, and directed/mixed structures. It develops constructive characterizations and orientation results, linking partition-connectivity with the existence of multiple disjoint spanning trees, arborescences, and hypertrees, and it discusses algorithmic implications via matroid sums and common-basis problems. The discussion extends to hypergraphs, dypergraphs, and rigidity theory, showing how high connectivity underpins generic rigidity and how recent results sharpen our understanding of partition-deficiency and equitable decompositions. The work highlights interdisciplinary connections, including rigidity, optimization, and game-theoretic switching, underscoring the enduring influence of the tree-packing/covering paradigm in combinatorics and beyond.

Abstract

One of the major starting points of discrete optimization is the theorem of Nash-Williams and Tutte on the existence of disjoint spanning trees of a graph along with its counterpart on the existence of forests covering all edges of the graph. These elegant results triggered a comprehensive research that gave rise to far-reaching generalizations and found applications at seemingly far-fetched areas. There are well over a thousand papers in the literature, including quite a few brand-new ones. Our first goal is to enlighten some aspects and links of these developments with the hope that the melody finds its way to non-experts. But we hope that experts will also find some novelties in our orchestration.

Paper Structure

This paper contains 14 sections, 35 theorems, 17 equations.

Table of Contents

  1. Introduction
  2. Some graph problems triggered by the tree-packing and tree-covering theorems
  3. Matroids in the background
  4. Some consequences on graphs
  5. Hypergraphs and directed graphs
  6. Hypergraphs
  7. Directed and mixed graphs
  8. Dypergraphs
  9. Orienting graphs and hypergraphs
  10. Graphs
  11. Hypegraphs
  12. Constructive characterizations
  13. Tree-connectivity in rigidity theory
  14. Switching games

Key Result

THEOREM 1.1

An undirected graph $G=(V,E)$ includes $k$ edge-disjoint spanning trees if and only if holds for every subset $F\subseteq E$ of edges, where $q(G,F)$ denotes the number of connected components of the subgraph of $G$ induced by $E-F$.

Theorems & Definitions (35)

  • THEOREM 1.1: Tutte Tutte61a, Theorem I
  • THEOREM 1.2: Nash-Williams Nash61, Theorem 1
  • THEOREM 1.3: Tree-packing theorem of Tutte and Nash-Williams
  • THEOREM 1.4: Tree-covering theorem of Nash-Williams Nash64
  • THEOREM 1.5: Horn Horn55
  • THEOREM 2.1: Tutte Tutte61a, Theorem II.
  • THEOREM 2.2: Edmonds Edmonds65a
  • THEOREM 2.3: Edmonds and Fulkerson Edmonds-Fulkerson, Theorem 2c
  • THEOREM 2.4: Edmonds and Fulkerson Edmonds-Fulkerson, Theorem 1c
  • THEOREM 2.5: Matroid-sum theorem of Edmonds Edmonds68, Theorems 1 and 2
  • ...and 25 more