Independent Roman {2}-domination in graphs

For a graph G=(V,E), a Roman {2}-dominating function (R2DF) f:V→{0,1,2} has the property that for every vertex v∈V with f(v)=0, either there exists an adjacent vertex, a neighbor u∈N(v), with f(u)=2, or at least two neighbors x,y∈N(v) having f(x)=f(y)=1. The weight of a R2DF is the sum f(V)=∑v∈Vf(v). A R2DF f=(V0,V1,V2) is called independent if V1∪V2 is an independent set. The independent Roman {2}-domination number i{R2}(G) is the minimum weight of an IR2DF on G. In this paper, we show that the decision problem associated with i{R2}(G) is NP-complete even when restricted to bipartite graphs. Then we show that for every graph G of order n, 0≤ir2(G)−i{R2}(G)≤n∕5 and 0≤iR(G)−i{R2}(G)≤n∕4, where ir2(G) and iR(G) are the independent 2-rainbow domination and independent Roman domination numbers, respectively. Moreover, we prove that the equality i{R2}(G)=ir2(G) holds for trees.