The [k]-multiple Roman domination in Graphs

Abstract In 2016, Beeler et al. introduced the double Roman domination as a variation of Roman domination and recently Abdollahzadeh et al. (resp. Amjadi et al.) initiated the study of triple (resp. quadruple) Roman domination in graphs. Given any labeling of the vertices of a graph, AN(v) stands for the set of neighbors of a vertex v having a positive label. In this paper we generalize these concepts by studying the [k]-multiple Roman domination functions ([k]-MRDF) in graphs which coincides with the previous versions when 2 ≤ k ≤ 4. Namely, f is a [k]-MRDF if f(N[v]) ≥ k + |AN(v)| for all v. We approach the complexity of the associate decision problem and we also give sharp bounds and exact values for several classes of graphs.