On graphs with equal domination and total domination numbers

Abstract

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.