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

TL;DR

This work exactly determines the dominion $\zeta(G_n)$ for the prism family $G_n=C_n\square P_2$ by encoding dominating sets as cyclic words over a four-letter alphabet and enforcing local constraints. The authors establish three distinct regimes by modular arithmetic: a rigid regime with $\zeta(G_n)=4$ for $n\equiv0\pmod{4}$, a linear regime with $\zeta(G_n)=2n$ for $n\equiv1,3\pmod{4}$, and a quadratic regime with $\zeta(G_n)=n(n+2)$ for $n\equiv2\pmod{4}$ (with $n\ge10$), plus exceptional values $\zeta(G_3)=9$ and $\zeta(G_6)=51$. The analysis combines forcing/exclusion rules, backbone/anchor decompositions, and an orbit–anchor counting mechanism to obtain exact counts, while also introducing normalized and composite robustness indices (e.g., $\eta$, $\mathcal{E}$, $\rho$, $\text{SFI}$, $\text{RRI}$, $\text{LDI}$) to quantify redundancy, resilience, and distribution of minimum dominators. The results reveal how dominion can exhibit markedly different growth patterns from the domination number, motivating extensions to other Cartesian products and the development of multivariate generating-function approaches for finer combinatorial control. Overall, the paper advances the quantitative understanding of domination structure in parametric graph families and offers a framework for assessing robustness beyond the minimum-size dominating sets.

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.

Figures (1)

• Figure 1: The prism graph $G_{10}=C_{10}\square P_{2}$ with vertex labels $t_i,b_i$.

Theorems & Definitions (36)

• Theorem 1.1: grinstead-slater-1991
• Lemma 2.1: Local domination constraints
• proof
• Theorem 3.1: Exact dominion of the prism
• Lemma 3.2: $\mathrm{C}\mathrm{C}$ forces doubles
• proof
• Lemma 3.3: An empty column forces complementary singleton neighbors
• proof
• Lemma 3.4: No $\mathrm{D}$ in minimum words
• proof