Quasiperfect domination in triangular lattices

TL;DR

This paper investigates perfect and quasiperfect dominating sets in the triangular lattice graph, classifying their structures and identifying conditions for their components in regular tessellations and toroidal quotients.

Contribution

It provides a classification of perfect and quasiperfect dominating sets in the triangular lattice and its toroidal quotients, detailing their component structures.

Findings

Classification of perfect dominating sets in the triangular lattice.

Most quasiperfect dominating sets have components of size 1, 2, or 3.

Results applicable to regular tessellations and toroidal quotients.

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 $S$.