  Let $G$ be a graph each component of which has order at least 3, and let $G$
have order $n$, size $m$, total domination number $\gamma_t$ and maximum degree
$\Delta(G)$. Let $\Delta = 3$ if $\Delta(G) = 2$ and $\Delta = \Delta (G)$ if
$\Delta(G) \ge 3$. It is known [J. Graph Theory 49 (2005), 285--290; J. Graph
Theory 54 (2007), 350--353] that $m \le \Delta (n- \gamma_t)$. In this paper we
characterize the extremal graphs $G$ satisfying $m = \Delta (n- \gamma_t)$.
