We study the domination number $γ(Q_n^3)$ 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 $γ(Q_n^3) = Θ(n^2)$. We also certify exact values for $n \leq 6$ via integer linear programming and independent verification.

翻译：我们研究三维$n \times n \times n$皇后图的控制数$\gamma(Q_n^3)$。主要结果是一个分层定理，该定理针对每种位置类型——角点、边点、面点或内部点——计算被一个皇后支配的内核顶点数，并特别表明内部放置比边界放置支配更多内核单元。这通过八面体群导出对称性简化原理，并补充标准计数下界与分层上界，得到$\gamma(Q_n^3) = \Theta(n^2)$。我们还通过整数线性规划及独立验证，确认了$n \leq 6$时的精确值。