发表机构
Dhirubhai Ambani University（Dhirubhai Ambani大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对无孤立顶点图，解决了控制数为2时控制数与全控制数相等的结构刻画问题，按围长分类推导相关性质并应用于极端图族，形成完整可检验的结构转化词典。
AI 中文摘要
对于没有孤立顶点的图$G$，控制数$\u03b3(G) \u2264$全控制数$\u03b3_t(G) \u2264 2\u03b3(G)$。尽管达到$\u03b3(G)=\u03b3_t(G)$的图已被广泛研究，但最小非平凡情形$\u03b3(G)=2$的完整结构刻画仍悬而未决。我们按围长$g(G)$解决该情形：当$g(G) \u2260 3$时，证明$\u03b3_t(G)=2$迫使$G$为二分图，给出该等式的精确度数和准则，并在最小度$\u03b4(G) \u2265 2$的额外假设下，证明$\u03b3_t(G) \u2265 \{2,4\}$；当$g(G)=3$时，利用Golumbic的顶点乘操作结合秩2至5图的已知分类，完整列出满足$\u03b3(G)=\u03b3_t(G)=2$的图族。作为应用，证明Erdős和Rényi识别、Henning和Southey分类的无支配顶点直径2图这一极端族中，所有图的$\u03b3_t(G) \u2265 \{3,6\}$，故其中不会出现$\u03b3=\u03b3_t=2$的情况。这些结果共同给出将$\u03b3_t(G)=2$转化为具体可检验图论性质的完整结构词典。
英文摘要
For a graph $G$ without isolated vertices, $γ(G)\leγ_t(G)\le 2γ(G)$. While graphs attaining $γ(G)=γ_t(G)$ have been studied extensively, a complete structural description in the smallest nontrivial case $γ(G)=2$ has remained open. We resolve this case according to girth. When $g(G)\ne 3$, we show $γ_t(G)=2$ forces $G$ bipartite, give an exact degree-sum criterion for this equality, and show $γ_t(G)\in\{2,4\}$ under the additional hypothesis $δ(G)\ge 2$. When $g(G)=3$, we use Golumbic's vertex-multiplication operation together with known classifications of graphs of rank $2$ through $5$ to completely list the families satisfying $γ(G)=γ_t(G)=2$. As an application, we show that every graph in the extremal family of diameter-two, dominating-vertex-free graphs identified by Erdős and Rényi and classified by Henning and Southey satisfies $γ_t(G)\in\{3,6\}$, so $γ=γ_t=2$ never occurs there. Together these results give a full structural dictionary translating $γ_t(G)=2$ into concrete, checkable graph-theoretic properties.
Comments12 pages, 3 figures
