How to see the forest for the trees

TL;DR

This paper reviews the development and generalizations of fundamental theorems on disjoint spanning trees and forests in graphs, aiming to clarify their links and inspire both experts and non-experts in the field.

Contribution

It provides an overview of the extensive research on spanning trees and forests, highlighting connections and recent developments in the area.

Findings

Clarifies links between classical theorems and recent generalizations

Highlights applications across various fields

Identifies potential directions for future research

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.