Bounds for (strong) Roman $k$-dominations

TL;DR

This paper establishes sharp upper bounds for Roman and strong Roman k-domination numbers in connected graphs, generalizing previous results for specific cases and characterizing extremal graphs.

Contribution

It provides unified, extended bounds for Roman k-domination parameters and characterizes the graphs that attain these bounds.

Findings

Sharp upper bounds for Roman and strong Roman k-domination numbers.

Unified results extending previous bounds for k=2 and k=3.

Complete characterization of extremal graphs achieving these bounds.

Abstract

Motivated by resource defense models in networks, such as protecting territories with varying legion strengths, let $k \geq 2$ be an integer. Roman $k$-domination and strong Roman $k$-domination generalize Roman, double Roman, Italian, and double Italian domination to arbitrary number of legions. The main goal of this note is establishing sharp upper bounds for the Roman and strong Roman $k$-domination numbers of connected graphs. These bounds unify and extend prior results for $k=2$ and $k=3$. We also precisely characterize the graphs achieving these bounds.