20.500.12556/DiRROS-29629 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 Neznano 2026-06-01 14:55:24 2026-06-01 14:55:26 2026-06-02 03:48:28 0000-00-00 00:00:00 2026 0 0 27 str. iss. 5, [article no.] 382 Vol. 15 2026 0000-00-00 Zaloznikova Objavljeno NiDoloceno 0000-00-00 0000-00-00 2026-01-01 519.17 2075-1680 10.3390/axioms15050382 279071491 axioms-15-00382-v3.pdf axioms-15-00382-v3.pdf 1 C45D7681A115BE40BF5E8434A5F0B747 7eb7364786fa72b144929f403552a88527c8308f0e5940bcf223d9aa4a91c66c 9a497d7e-5db9-11f1-8e51-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=44498 https://www.mdpi.com/2075-1680/15/5/382 1 2c7d7e4d-5db9-11f1-8e51-001a4af901a5 https://dirros.openscience.si/Dokument.php?lang=slv&id=44497 Inštitut za matematiko, fiziko in mehaniko Rudolfovo – Znanstveno in tehnološko središče Novo mesto 0 0 0