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 $G\setminus S$ 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 $S$.

💡 Research Summary

The paper investigates two refined domination concepts—perfect domination and quasiperfect domination—on the regular triangular lattice (the planar tessellation with Schlӓfli symbol {3,6}) and on its finite toroidal quotients. A vertex subset S of a graph G is called a perfect dominating set if every vertex outside S has exactly one neighbor in S; it is called a quasiperfect dominating set if each external vertex has either one or two neighbors in S. These definitions tighten the classic domination notion by limiting the number of dominating neighbors, which yields highly regular and constrained configurations.

The authors first focus on the infinite triangular lattice, a 6‑regular planar graph where each vertex participates in six equilateral triangles. By exploiting the lattice’s translational symmetry, they decompose the plane into fundamental cells (typically 2 × 2 or 3 × 3 blocks) and enumerate all possible local placements of S inside a cell that satisfy the domination constraints. They then impose compatibility conditions on adjacent cells, which can be expressed through a transition matrix or a set of “color‑matching” rules. This systematic local‑to‑global approach leads to a complete classification of perfect dominating sets: only two periodic patterns survive.

The first pattern is a chessboard‑type 2‑coloring: vertices are selected on one of the two bipartite classes of the lattice, producing a checkerboard where each non‑selected vertex touches exactly one selected neighbor. This pattern can be realized on a torus only when both dimensions m and n are even, because the periodic boundary must preserve the bipartite alternation. The second pattern is a “triangular‑spacing” configuration, where a selected vertex appears every three steps along each lattice direction, forming a sparser sublattice of index three. This pattern requires both m and n to be multiples of three. No other arrangements satisfy the strict one‑neighbor condition, and the authors prove this by eliminating all alternative local configurations through exhaustive case analysis.

Turning to quasiperfect domination, the paper relaxes the neighbor count to allow up to two dominating vertices per external vertex. To keep the analysis tractable, the authors restrict the induced subgraph G