Further results on \([k]\)-Roman domination on cylindrical grids \(C_m \Box P_n\)

TL;DR

This paper investigates the $[k]$-Roman domination number of cylindrical graphs, establishing bounds, characterizing optimal configurations, and providing explicit constructions for small fixed widths, with analysis on how bounds vary with parameters.

Contribution

It offers new bounds, characterizations, and explicit constructions for the $[k]$-Roman domination number on cylindrical grids, advancing understanding of domination parameters in these graphs.

Findings

Lower bound: $ ext{γ}_{[k]R}(C_mox P_n) > (k+1)\lceilrac{mn}{5} ceil$

Explicit periodic dominating functions for $m=5, o,8$

Bounds depend on $k$ and path length, with comparative analysis

Abstract

In this paper, we study the $[k]$-Roman domination number of cylindrical graphs $C_m \Box P_n$. Our analysis begins with a general lower bound based on local neighborhood constraints, showing that $\gamma_{[k]R}(C_m\Box P_n) > (k+1)\left\lceil\frac{mn}{5}\right\rceil.$ By exploiting the connection between $[k]$-Roman domination and efficient domination, we characterize those cylindrical graphs whose optimal $[k]$-Roman domination number is realized by configurations with minimum possible local neighborhood weight. For fixed small values $m\in\{5,\ldots,8\}$, we construct explicit periodic $[k]$-Roman dominating functions that yield constructive upper bounds. These constructions are further refined using ceiling-type adjustments and reductions based on packing sets. A systematic comparison of the resulting bounds shows how their relative strength depends on the parameter $k$ and on the length of the path.