Exact Dominion of the Prism Graph: Enumeration by Congruence Class via Cyclic Words

Let G n = C n □ P 2 be the prism graph on 2 n vertices. The dominion ζ ( G n ) counts the minimum dominating sets of G n . Encoding column selections as cyclic words over a quaternary alphabet converts domination into explicit local adjacency constraints, reducing the count of minimum dominating sets to the enumeration of minimum‐weight admissible words. This yields closed‐form formulas for ζ ( G n ) for all n ≥ 3, stratified by n mod 4, with the exceptional cases n ∈ {3, 6} confirmed by exhaustive computation.