Note on 2-rainbow domination and Roman domination in graphs

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