Quasiperfect domination in triangular lattices

A vertex subset S of a graph G is a perfect (resp. quasiperfect) dominating set in G if each vertex v of G∖S is adjacent to only one vertex ($d_v$ ∈ {1,2} vertices) of S. Perfect and quasiperfect dominating sets in the regular tessellation graph of Schläfli 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 $K_ν$, where ν ∈ {1,2,3} depends only on S.