On the Structure of 3D Queen Domination

TL;DR

This paper analyzes the domination number of 3D queen graphs, providing a stratified theorem for different position types, symmetry reduction, bounds, and exact values for small n.

Contribution

It introduces a stratified approach for computing domination numbers in 3D queen graphs and establishes bounds and exact values for small cases.

Findings

Interior placements dominate more core cells than boundary placements.

The domination number scales as Theta(n^2).

Exact values verified for n ≤ 6.

Abstract

We study the domination number $\gamma(Q_n^3)$ of the three-dimensional $n \times n \times n$ queen graph. The main result is a stratified theorem computing, for each position type -- corner, edge, face, or interior -- the number of inner-core vertices dominated by a queen, and showing in particular that interior placements dominate strictly more core cells than boundary placements. This yields a symmetry-reduction principle via the octahedral group and complements the standard counting lower bound and layered upper bound, giving $\gamma(Q_n^3) = \Theta(n^2)$. We also certify exact values for $n \leq 6$ via integer linear programming and independent verification.