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

Let G_n = C_n square P_2 denote the prism (circular ladder) graph on 2n vertices. By encoding column configurations as cyclic words, domination is reduced to local Boolean constraints on adjacent factors. This framework yields explicit formulas for the dominion zeta(G_n), stratified by n mod 4, with the exceptional cases n in {3, 6} confirmed computationally. Together with the known domination numbers gamma(G_n), these results expose distinct arithmetic regimes governing optimal domination, ranging from rigid forcing to substantial enumerative flexibility, and motivate quantitative parameters for assessing structural robustness in parametric graph families.