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 $S$.
Categories: math.CO cs.IT math.IT