On the Structure of 3D Queen Domination

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.