2-switch: transition and stability on forests and pseudofests

TL;DR

This paper demonstrates that any two forests or pseudoforests with identical degree sequences can be transformed into each other through a sequence of 2-switches while maintaining the forest or pseudoforest property, and explores the implications for graph parameters.

Contribution

It introduces a method to transform forests and pseudoforests with the same degree sequence via 2-switches, preserving their structure and analyzing parameter perturbations.

Findings

Any two forests or pseudoforests with the same degree sequence can be connected through 2-switches.

The 2-switch operation minimally perturbs certain graph parameters.

Parameters exhibit the interval property within these graph families.

Abstract

Given any two forests (pseudoforests) with the same degree sequence, we show that one can be transformed into the other by a sequence of 2-switches in such a way that all the intermediate graphs of the transformation are forests (pseudoforests). We also prove that the 2-switch operation perturbs minimally some well-known integer parameters in families of graphs with the same degree sequence. Then, we apply these results to conclude that the studied parameters have the interval property on those families.