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

TL;DR

This paper develops a combinatorial framework to exactly enumerate the domination properties of prism graphs using cyclic words, revealing arithmetic regimes and structural robustness in these graph families.

Contribution

It introduces a novel encoding of domination problems via cyclic words, providing explicit formulas and computational confirmation for the prism graph's domination zeta function.

Findings

Explicit formulas for domination zeta(G_n) stratified by n mod 4

Identification of distinct arithmetic regimes in domination behavior

Computational confirmation for exceptional cases n=3,6

Abstract

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.