%0 Journal Article 
%A Valenzuela Tripodoro, Juan Carlos
%A Mateos Camacho, María Antonia
%A Cera López, Martín
%A Álvarez Ruiz, María del Pilar
%T Further Results on the [k]-Roman Domination in Graphs
%D 2024
%@ 1735-8515
%U http://hdl.handle.net/10498/33191
%X In 2016, Beeler et al. defined the double Roman domination as a variation of Roman
domination. Sometime later, in 2021, Ahangar et al. introduced the concept of [k]-
Roman domination in graphs and settled some results on the triple Roman domination
case. In 2022, Amjadi et al. studied the quadruple version of this Roman-domination-
type problem. 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 continue the study
of the [k]-Roman domination functions ([k]-RDF) in graphs which coincides with the
previous versions when 2 ≤ k ≤ 4. Namely, f is a [k]-RDF if f (N [v]) ≥ k +|AN (v)|
for all v. We prove that the associate decision problem is NP-complete even when
restricted to star convex and comb convex bipartite graphs and we also give sharp
bounds and exact values for several classes of graphs
%K Roman domination
%K Double Roman domination
%K Triple Roman domination
%K Quadruple Roman domination
%~ Universidad de Cádiz