  A total dominating set of a graph $G=(V,E)$ is a subset $D$ of $V$ such that
every vertex in $V$ is adjacent to at least one vertex in $D$. The total
domination number of $G$, denoted by $\gamma _t (G)$, is the minimum
cardinality of a total dominating set of $G$. A near-triangulation is a
biconnected planar graph that admits a plane embedding such that all of its
faces are triangles except possibly the outer face. We show in this paper that
$\gamma _t (G) \le \lfloor \frac{2n}{5}\rfloor$ for any near-triangulation $G$
of order $n\ge 5$, with two exceptions.
