Quasiperfect domination in triangular lattices

Abstract: 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 $S$.