$[k]$-Roman domination on cylindrical grids $C_m \Box P_n$

TL;DR

This paper studies the $[k]$-Roman domination number on cylindrical grids, providing bounds, explicit patterns, and structural insights that unify various domination parameters and reveal new regularities with increasing reinforcement strength.

Contribution

It introduces bounds and explicit labeling patterns for $[k]$-Roman domination on cylindrical grids, extending and unifying existing domination parameters with structural analysis.

Findings

Derived explicit bounds for $ ext{γ}_{[k]R}(C_m imes P_n)$.

Identified conditions for efficient dominating sets in cylindrical graphs.

Presented exact packing numbers complementing domination results.

Abstract

Roman domination and its higher-order extensions have attracted considerable attention due to their natural interpretation in terms of defensive resource allocation on networks. The recently introduced $[k]$-Roman domination framework unifies classical Roman, double, triple, and higher-strength protection schemes by allowing each fortified vertex to provide up to $k$ levels of support. In this paper, we investigate the $[k]$-Roman domination number $\gamma_{[k]R}(G)$ on cylindrical grids $C_m \Box P_n$. We relate $[k]$-Roman domination to efficient domination and show that for efficient graphs one has $\gamma_{[k]R}(G)=(k+1)\gamma(G)$; as a consequence, we obtain explicit values for broad families of toroidal grids and determine exactly when the cylindrical graphs $C_m\Box P_n$ admit an efficient dominating set. Building on these structural insights, we derive several upper bounds for $\gamma_{[k]R}(C_m \Box P_n)$ for small fixed values of $m$, accompanied by explicit labeling patterns that attain these bounds. All obtained bounds are systematically compared, revealing parameter ranges in which different constructions dominate depending on the value of $k$ and the length of the path. In addition, we present exact packing numbers for selected cylindrical graphs, which complement the domination results and enable further refinements via local weight reductions. Our results extend and unify known domination-type parameters on grid-like structures and highlight new regularities that emerge as the reinforcement strength increases.