Upper bounds for double Roman domination and $[k]$-Roman domination of cylindrical graphs $C_m \Box P_n$

Brezovnik, Simon (author), Žerovnik, Janez (author)

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.

• Language: English
• Keywords: $[k]$-Roman domination, double Roman domination, cylindrical grids, Cartesian product of graphs
• Work type: Article
• Typology: 1.01 - Original Scientific Article
• Organization: FS - Faculty of Mechanical Engineering; FMF - Faculty of Mathematics and Physics
• Publication status: Published
• Publication version: Version of Record
• Year: 2026
• Number of pages: 27 str.
• Numbering: Vol. 15, issue 5, art. 382
• PID: 20.500.12556/RUL-182830
• UDC: 519.17
• ISSN on article: 2075-1680
• DOI: 10.3390/axioms15050382
• COBISS.SI-ID: 279071491
• Publication date in RUL: 25.05.2026

Record is a part of a journal

• Title: Axioms
• Shortened title: Axioms
• Publisher: MDPI
• ISSN: 2075-1680
• COBISS.SI-ID: 519951897

Licences

• License: CC BY 4.0, Creative Commons Attribution 4.0 International
• Link: http://creativecommons.org/licenses/by/4.0/
• Description: This is the standard Creative Commons license that gives others maximum freedom to do what they want with the work as long as they credit the author.

Secondary language

• Language: Slovenian
• Keywords: $[k]$-rimska dominacija, dvojna rimska dominacija, cilindrični grafi, kartezični produkt grafov