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On the Structure of 3D Queen Domination

Mahesh Ramani

Abstract

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.

On the Structure of 3D Queen Domination

Abstract

We study the domination number of the three-dimensional 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 . We also certify exact values for via integer linear programming and independent verification.

Paper Structure

This paper contains 8 sections, 6 theorems, 11 equations, 1 table.

Table of Contents

  1. Introduction
  2. Core Coverage by Position Type
  3. Bounds
  4. Computational Certification
  5. The domination ILP
  6. Automorphisms of $Q^3_n$ and the fundamental domain
  7. Decomposition by first-queen position
  8. Independent verification

Key Result

Lemma 1

Let $M \geq 1$ be an integer and let $f(a,c) = |a-c| + |a+c-M|$ for integers $0 \leq a, c \leq M$. Then $f(a,c) \equiv M \pmod{2}$ for all $a, c$. Hence $f(a,c) \geq M \bmod 2$. If $M$ is even, equality holds if and only if $a = c = M/2$. If $M$ is odd, equality holds exactly at $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (12)

  • Lemma 1
  • proof
  • Theorem 2
  • proof
  • Corollary 3
  • proof
  • Theorem 4: barrrao2007
  • proof
  • Theorem 5
  • proof
  • ...and 2 more