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Further results on \([k]\)-Roman domination on cylindrical grids \(C_m \Box P_n\)

Simon Brezovnik, Janez Žerovnik

Abstract

In this paper, we study the $[k]$-Roman domination number of cylindrical graphs $C_m \Box P_n$. Our analysis begins with a general lower bound based on local neighborhood constraints, showing that $γ_{[k]R}(C_m\Box P_n) > (k+1)\left\lceil\frac{mn}{5}\right\rceil.$ By exploiting the connection between $[k]$-Roman domination and efficient domination, we characterize those cylindrical graphs whose optimal $[k]$-Roman domination number is realized by configurations with minimum possible local neighborhood weight. For fixed small values $m\in\{5,\ldots,8\}$, we construct explicit periodic $[k]$-Roman dominating functions that yield constructive upper bounds. These constructions are further refined using ceiling-type adjustments and reductions based on packing sets. A systematic comparison of the resulting bounds shows how their relative strength depends on the parameter $k$ and on the length of the path.

Further results on \([k]\)-Roman domination on cylindrical grids \(C_m \Box P_n\)

Abstract

In this paper, we study the -Roman domination number of cylindrical graphs . Our analysis begins with a general lower bound based on local neighborhood constraints, showing that By exploiting the connection between -Roman domination and efficient domination, we characterize those cylindrical graphs whose optimal -Roman domination number is realized by configurations with minimum possible local neighborhood weight. For fixed small values , we construct explicit periodic -Roman dominating functions that yield constructive upper bounds. These constructions are further refined using ceiling-type adjustments and reductions based on packing sets. A systematic comparison of the resulting bounds shows how their relative strength depends on the parameter and on the length of the path.

Paper Structure

This paper contains 9 sections, 18 theorems, 81 equations, 10 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lower bound for $\gamma_{[k]R}(C_m \Box P_n)$
  4. Upper bounds for $\gamma_{[k]R}(C_m\Box P_n)$
  5. Upper bounds for $m=5$
  6. Upper bounds for $m=6$
  7. Upper bounds for $m=7$
  8. Upper bounds for $m=8$
  9. Conclusion

Key Result

Proposition 1

If $k \ge 2$, then in a $\gamma_{[k]R}(G)$-RDF, no vertex is assigned the value $1$.

Figures (10)

  • Figure 1: A periodic packing pattern in $C_5\Box P_n$. The stars indicate the vertices selected in the packing set in each fibre.
  • Figure 2: Best bound among Bounds \ref{['eq:ub-C5Pn-linear']}, \ref{['U']}, and \ref{['B']} for $4\le n\le20$ and $1\le k\le22$.
  • Figure 3: A periodic packing pattern in $C_6\Box P_n$. In every second fibre we choose two vertices at distance $3$ in the $C_6$-direction, and alternate the pair by a shift of $1$.
  • Figure 4: Best bound among Bounds \ref{['dve']}, \ref{['U_2']}, and \ref{['B_2']} for $n=4,\dots,20$ and $k=1,2,\dots,25$ for $C_6\Box P_n$.
  • Figure 5: Best bound among Bounds \ref{['dve']}, \ref{['U_2']}, and \ref{['B_2']} for $n=90,\dots,110$ and $k=40,\dots,57$ for $C_6\Box P_n$.
  • ...and 5 more figures

Theorems & Definitions (39)

  • Proposition 1: Khalili2023
  • Theorem 2
  • proof
  • Theorem 4
  • proof
  • Theorem 5
  • proof
  • proof
  • Theorem 7
  • proof
  • ...and 29 more