  A vertex subset $S$ of a graph $G$ is a perfect (resp. quasiperfect)
dominating set in $G$ if each vertex $v$ of $G\setminus S$ is adjacent to only
one vertex ($d_v\in\{1,2\}$ vertices) of $S$. Perfect and quasiperfect
dominating sets in the regular tessellation graph of Schl\"afli symbol
$\{3,6\}$ and in its toroidal quotients are investigated, yielding the
classification of their perfect dominating sets and most of their quasiperfect
dominating sets $S$ with induced components of the form $K_{\nu}$, where
$\nu\in\{1,2,3\}$ depends only on $S$.
