(Independent) Roman {2}-Domination Number on Graphs

For a graph [Formula: see text], a function [Formula: see text] is a Roman {2}-dominating function (R{2}DF) if [Formula: see text] satisfies the following condition: if [Formula: see text] then there is [Formula: see text] such that [Formula: see text] or there is [Formula: see text] such that [Formula: see text]. Let [Formula: see text] denote the set of vertices assigned [Formula: see text] by function [Formula: see text]. The weight of a R{2}DF [Formula: see text] is the sum [Formula: see text]. The Roman {2}-domination number is the minimum weight of a R{2}DF on [Formula: see text], denoted by [Formula: see text]. A function [Formula: see text] is an independent Roman {2}-dominating function (IR{2}DF) if [Formula: see text] is a R{2}DF and [Formula: see text] is an independent set. In this paper, we initially study the influence of edge addition on the Roman {2}-domination number, and for two positive integer [Formula: see text] and [Formula: see text], we construct a graph [Formula: see text] and an induced subgraph [Formula: see text] of [Formula: see text] such that [Formula: see text] and [Formula: see text]. Based on this construction, we conclude that there is no relation between the Roman {2}-domination number of a graph and its induced subgraph. Furthermore, we give a bound for the independent Roman [Formula: see text]-domination number on proper interval graphs, and a linear-time algorithm for computing the independent Roman [Formula: see text]-domination number of unicyclic graphs with restrictions.