Classification of prime graphs with 2-switch-degree at most 4

The 2-switch-degree $\text{deg}(G)$ of a graph $G$ is the number of 2-switches that can be performed on $G$; equivalently, it is the degree of $G$ as a vertex of the realization graph $\mathcal{G}(d)$ of its degree sequence $d$. We classify the prime graphs of 2-switch-degree at most 4, where a grap