发表机构
Instituto de Matemática Aplicada San Luis (IMASL), UNSL–CONICET（圣路易斯应用数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文分类了2-开关度至多4的素图，并由此得出一个尖锐二分类：当实现度k≤3时，实现图顶点传递且k-正则当且仅当不含T_221及其补图作为诱导子图；且对每个k≥4存在度k的素图可提升度至2k-2，最终仅10个实现图。
AI 中文摘要
图$G$的2-开关度$\text{deg}(G)$是可对$G$执行的2-开关的数量；等价地，它是$G$作为其度序列$d$的实现图$\mathcal{G}(d)$的顶点的度。我们对2-开关度至多为4的素图进行分类，其中若一个图关于Tyshkevich合成是不可分解的，且每个顶点都参与某个2-开关，则该图是素的。从这一分类中，我们推导出一个尖锐的二分类，它从$\mathcal{G}(d)$的单个局部度恢复其全局形状：若$d$有一个实现$X$且$\text{deg}(X)=k\le3$，则$\mathcal{G}(d)$是顶点传递的且$k$-正则的，当且仅当$T_{221}$和$\overline{T_{221}}$都不是$X$的诱导子图。此外，对于每个$k\geq 4$，存在某个度为$k$的素图，其携带一个2-开关将其度提升至$2k-2$。作为进一步的结果，在同构意义下，度至多为4的素图的实现图仅有10个。
英文摘要
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 graph is prime if it is indecomposable with respect to the Tyshkevich composition and every vertex takes part in some 2-switch. From this classification we derive a sharp dichotomy that recovers the global shape of $\mathcal{G}(d)$ from a single one of its local degrees: if $d$ has a realization $X$ with $\text{deg}(X)=k\le3$, then $\mathcal{G}(d)$ is vertex-transitive and $k$-regular if and only if neither $T_{221}$ nor $\overline{T_{221}}$ is an induced subgraph of $X$. Moreover, for every $k\geq 4$ some prime graph of degree $k$ carries a 2-switch raising its degree to $2k-2$. As a further consequence, up to isomorphism there are only 10 realization graphs of prime graphs with degree at most 4.
