Difference between 22-rainbow domination and Roman domination in graphs

A 22-rainbow dominating function  ff of a graph GG is a function f:V(G)→2{1,2}f:V(G)→2{1,2} such that, for each vertex v∈V(G)v∈V(G) with f(v)=0̸f(v)=0̸, ⋃u∈NG(v)f(u)={1,2}⋃u∈NG(v)f(u)={1,2}. The minimum of ∑v∈V(G)|f(v)|∑v∈V(G)|f(v)| over all such functions is called the 22-rainbow domination number  γr2(G)γr2(G). A Roman dominating function  gg of a graph GG, is a function g:V(G)→{0,1,2}g:V(G)→{0,1,2} such that, for each vertex v∈V(G)v∈V(G) with g(v)=0g(v)=0, vv is adjacent to a vertex uu with g(u)=2g(u)=2. The minimum of ∑v∈V(G)g(v)∑v∈V(G)g(v) over all such functions is called the Roman domination number  γR(G)γR(G).Regarding 0̸0̸ as 00, these two dominating functions have a common property that the same three integers are used and a vertex having 00 must be adjacent to a vertex having 22. Motivated by this similarity, we study the relationship between γR(G)γR(G) and γr2(G)γr2(G). We also give some sharp upper bounds on these dominating functions. Moreover, one of our results tells us the following general property in connected graphs: any connected graph GG of order n≥3n≥3 contains a bipartite subgraph H=(A,B)H=(A,B) such that δ(H)≥1δ(H)≥1 and |A|−|B|≥n/5|A|−|B|≥n/5. The bound on |A|−|B||A|−|B| is best possible.