On graphs with equal domination and total domination numbers
For a graph $G$ without isolated vertices, $\gamma(G)\le\gamma_t(G)\le 2\gamma(G)$. While graphs attaining $\gamma(G)=\gamma_t(G)$ have been studied extensively, a complete structural description in the smallest nontrivial case $\gamma(G)=2$ has remained open. We resolve this case according to girth. When $g(G)\ne 3$, we show $\gamma_t(G)=2$ forces $G$ bipartite, give an exact degree-sum criterion for this equality, and show $\gamma_t(G)\in\{2,4\}$ under the additional hypothesis $\delta(G)\ge 2$. 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 $\gamma(G)=\gamma_t(G)=2$. As an application, we show that every graph in the extremal family of diameter-two, dominating-vertex-free graphs identified by Erd\H{o}s and R\'enyi and classified by Henning and Southey satisfies $\gamma_t(G)\in\{3,6\}$, so $\gamma=\gamma_t=2$ never occurs there. Together these results give a full structural dictionary translating $\gamma_t(G)=2$ into concrete, checkable graph-theoretic properties.