The 2-switch-degree of a graph

TL;DR

This paper investigates the 2-switch-degree of a graph, analyzing its properties, formulas, and behavior in specific graph families to deepen understanding of graph realization spaces.

Contribution

It introduces explicit formulas and characterizes the 2-switch-degree for various graph families, advancing the theoretical understanding of graph realization graphs.

Findings

Derived explicit formulas for 2-switch-degree

Characterized active and inactive vertices

Analyzed behavior in trees and unicyclic graphs

Abstract

In this work, we delve into the study of the 2-switch-degree of a graph $G$, which is nothing more than the degree of $G$ as a vertex of the realization graph $\mathcal{G}(d)$ associated with the degree sequence $d$ of $G$. We explore the characteristics of active and inactive vertices, the basic properties of the degree, explicit formulas for its computation, and its behavior in specific families of graphs, such as trees and unicyclic graphs.