| Title: | Upper bounds for double Roman domination and $[k]$-Roman domination of cylindrical graphs $C_m\Box P_n$ |
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| Authors: | ID Brezovnik, Simon (Author)
ID Žerovnik, Janez (Author) |
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| Files: |  PDF - Presentation file, download (857,23 KB)
MD5: C45D7681A115BE40BF5E8434A5F0B747
 URL - Source URL, visit https://www.mdpi.com/2075-1680/15/5/382
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| Language: | English |
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| Typology: | 1.01 - Original Scientific Article |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
 RUDOLFOVO - Rudolfovo - Science and Technology Centre Novo Mesto
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| Abstract: | 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. |
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| Keywords: | [k]-Roman domination, double Roman domination, cylindrical grids, Cartesian product of graphs |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Publication date: | 01.01.2026 |
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| Year of publishing: | 2026 |
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| Number of pages: | 27 str. |
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| Numbering: | Vol. 15, iss. 5, [article no.] 382 |
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| PID: | 20.500.12556/DiRROS-29629  |
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| UDC: | 519.17 |
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| ISSN on article: | 2075-1680 |
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| DOI: | 10.3390/axioms15050382 |
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| COBISS.SI-ID: | 279071491 |
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| Note: |
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| Publication date in DiRROS: | 01.06.2026 |
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| Views: | 38 |
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| Downloads: | 21 |
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