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 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 $K_{\nu }$, where $\nu \in \{1,2,3\}$ depends only on $S$.

20 pages, 9 figures, 5 arrays