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 (dv ∈ {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 Kν , where ν ∈ {1, 2, 3} depends only on S.