Note on 2-rainbow domination and Roman domination in graphs

A Roman dominating function   of a graph GG is a function f:V→{0,1,2}f:V→{0,1,2} such that every vertex with 0 has a neighbor with 2. The minimum of f(V(G))=∑v∈Vf(v)f(V(G))=∑v∈Vf(v) over all such functions is called the Roman domination number  γR(G)γR(G). A 2-rainbow dominating function   of a graph GG is a function gg that assigns to each vertex a set of colors chosen from the set {1,2}{1,2}, for each vertex v∈V(G)v∈V(G) such that g(v)=0̸g(v)=0̸, we have ⋃u∈N(v)g(u)={1,2}⋃u∈N(v)g(u)={1,2}. The 2-rainbow domination number  γr2(G)γr2(G) is the minimum of w(g)=∑v∈V|g(v)|w(g)=∑v∈V|g(v)| over all such functions. We prove γr2(G)≤γR(G)γr2(G)≤γR(G) and obtain sharp lower and upper bounds for γr2(G)+γr2(G¯). We also show that for any connected graph GG of order n≥3n≥3, γr2(G)+γ(G)2≤n. Finally, we give a proof of the characterization of graphs with γR(G)=γ(G)+kγR(G)=γ(G)+k for 2≤k≤γ(G)2≤k≤γ(G).