Upper bounds for double Roman domination and $[k]$-Roman domination of cylindrical graphs $C_m \Box P_n$
Fuente:
arXiv
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866908960607436800 |
|---|---|
| author | Brezovnik, Simon Žerovnik, Janez |
| author_facet | Brezovnik, Simon Žerovnik, Janez |
| contents | 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 $3,\dots,9$, 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 $γ_{[2]R}(C_m\Box P_n)$ 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_12029 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Upper bounds for double Roman domination and $[k]$-Roman domination of cylindrical graphs $C_m \Box P_n$ Brezovnik, Simon Žerovnik, Janez Combinatorics 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 $3,\dots,9$, 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 $γ_{[2]R}(C_m\Box P_n)$ 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. |
| title | Upper bounds for double Roman domination and $[k]$-Roman domination of cylindrical graphs $C_m \Box P_n$ |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2604.12029 |