On a generalization of Roman domination with more legions.

In this paper, 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. [ABSTRACT FROM AUTHOR]