Grid Domination on Hexagonal Boards

Abstract

A grid on a cell of a game board attacks all neighboring cells. The domination number counts the minimum number of grids such that each cell of a board is occupied or attacked by a grid.

For square boards (chess boards), the domination number has been determined in a series of papers. Here, we start to consider grids on hexagon boards $B_n$ as parts of the Euclidean tessellation by congruent regular hexagons, where $B_1$ is one hexagon, $B_2$ consists of the three hexagons around one vertex, and $B_n$ for $n\ge 3$ consists of $B_{n-2}$ together with all hexagons having at least one hexagon in common with $B_{n-2}$.

An upper bound is presented for the grid domination number, and exact values are determined by computer for small $n$.