20.500.12556/RUL-182830 Upper bounds for double Roman domination and $[k]$-Roman domination of cylindrical graphs $C_m \Box P_n$ Roman-type domination parameters form an important class of graph invariants that model protection and resource allocation problems on networks. Among them, $[k]$-Roman domination provides a unified framework that generalizes Roman, double Roman, and higher-order variants. In this paper we investigate the $[k]$-Roman domination number of cylindrical grids $C_m\Box P_n$ and derive several new constructive upper bounds. Our approach combines three complementary techniques: linear periodic constructions, uniform ceiling-type labelings, and packing-based refinements. We first analyze the case $C_9\Box P_n$, where these three families of bounds can be compared explicitly and their relative efficiency is shown to depend on the parameter $k$. We then extend the linear constructions to cylindrical grids whose circumference is a multiple of one of the values $r \in 3,\dots,9$, obtaining a unified family of upper bounds for $C_{rt}\Box P_n$. Motivated by the asymptotic behavior of these estimates, we further derive general upper bounds depending only on the residue class of $m$ modulo $5$, which apply to all cylindrical grids. As a consequence, we obtain explicit estimates for the double Roman domination number $\gamma_{[2R]}(C_m\Box P_n)$ and compare the resulting multiple-based constructions with the residue-class bounds. This comparison shows that the residue-class construction becomes asymptotically superior for all sufficiently large admissible circumferences, while several exceptional small cases remain better covered by tailored constructions. $[k]$-Roman domination double Roman domination cylindrical grids Cartesian product of graphs $[k]$-rimska dominacija dvojna rimska dominacija cilindrični grafi kartezični produkt grafov true false true Angleški jezik Slovenski jezik Članek v reviji 2026-05-25 11:31:53 2026-05-25 11:31:59 2026-06-02 08:12:12 0000-00-00 00:00:00 2026 0 0 27 str. issue 5, art. 382 Vol. 15 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 0000-00-00 519.17 2075-1680 10.3390/axioms15050382 279071491 RAZ_Brezovnik_Simon_2026.pdf RAZ_Brezovnik_Simon_2026.pdf 1 702FA23FEC730DEA5A914507F84E442B f97800ee0897f27fdd35f2b369f518d1e1513f7306f6dc5bbc7fb54380daafd2 758441dc-581c-11f1-b0ab-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=234385 https://www.mdpi.com/2075-1680/15/5/382 1 3ed41eee-581c-11f1-b0ab-0050569b8976 https://repozitorij.uni-lj.si/Dokument.php?lang=slv&id=234384 Fakulteta za strojništvo Fakulteta za matematiko in fiziko 0 0 0