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

Brezovnik, Simon (avtor), Žerovnik, Janez (avtor)

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.

• Jezik: Angleški jezik
• Ključne besede: $[k]$-Roman domination, double Roman domination, cylindrical grids, Cartesian product of graphs
• Vrsta gradiva: Članek v reviji
• Tipologija: 1.01 - Izvirni znanstveni članek
• Organizacija: FS - Fakulteta za strojništvo ; FMF - Fakulteta za matematiko in fiziko
• Status publikacije: Objavljeno
• Različica publikacije: Objavljena publikacija
• Leto izida: 2026
• Št. strani: 27 str.
• Številčenje: Vol. 15, issue 5, art. 382
• PID: 20.500.12556/RUL-182830
• UDK: 519.17
• ISSN pri članku: 2075-1680
• DOI: 10.3390/axioms15050382
• COBISS.SI-ID: 279071491
• Datum objave v RUL: 25.05.2026

Gradivo je del revije

• Naslov: Axioms
• Skrajšan naslov: Axioms
• Založnik: MDPI
• ISSN: 2075-1680
• COBISS.SI-ID: 519951897

Licence

• Licenca: CC BY 4.0, Creative Commons Priznanje avtorstva 4.0 Mednarodna
• Povezava: http://creativecommons.org/licenses/by/4.0/deed.sl
• Opis: To je standardna licenca Creative Commons, ki daje uporabnikom največ možnosti za nadaljnjo uporabo dela, pri čemer morajo navesti avtorja.

Sekundarni jezik

• Jezik: Slovenski jezik
• Ključne besede: $[k]$-rimska dominacija, dvojna rimska dominacija, cilindrični grafi, kartezični produkt grafov