On a generalization of Roman domination with more legions

TL;DR

This paper introduces generalized Roman domination concepts, including strong and k-domination, and analyzes their domination numbers specifically in complete bipartite graphs, expanding the theoretical framework of Roman domination.

Contribution

It extends Roman domination theory by defining new variants like strong Roman and Roman k-domination, and provides initial results on their parameters in bipartite graphs.

Findings

Comparison of domination numbers across different Roman domination variants

Calculation of Roman k-domination numbers for complete bipartite graphs

Introduction of generalized Roman domination concepts

Abstract

In this note, we generalize the concepts of (perfect) Roman and Italian dominations to (perfect) strong Roman and Roman k-domination for arbitrary positive integer k. These generalizations cover some of previous ones. After some comparison of their domination numbers, as a first study of these concepts, we study the (perfect, strong, perfect strong) Roman k-domination numbers of complete bipartite graphs.