From spanning forests to edge subsets

TL;DR

This paper provides a new perspective on Tutte's concept of active edges by showing how any edge subset can be uniquely constructed from a spanning forest through specific edge modifications.

Contribution

It introduces a novel characterization of edge subsets in terms of unique internally and externally active edges associated with spanning forests.

Findings

Every edge subset can be uniquely constructed from a spanning forest.

The construction involves deleting internally active edges and adding externally active edges.

Provides insight into Tutte's definitions of active edges.

Abstract

We give some insight into Tutte's definition of internally and externally active edges for spanning forests. Namely we prove, that every edge subset can be constructed from the edges of exactly one spanning forest by deleting a unique subset of the internally active edges and adding a unique subset of the externally active edges.