A Note on Outer-Independent 2-Rainbow Domination in Graphs

Let G be a graph with vertex set V ( G ) and f : V ( G ) → { ∅ , { 1 } , { 2 } , { 1 , 2 } } be a function. We say that f is an outer-independent 2-rainbow dominating function on G if the following two conditions hold: ( i ) V ∅ = { x ∈ V ( G ) : f ( x ) = ∅ } is an independent set of G . ( ii ) ∪ u ∈ N ( v ) f ( u ) = { 1 , 2 } for every vertex v ∈ V ∅ . The outer-independent 2-rainbow domination number of G , denoted by γ r 2 o i ( G ) , is the minimum weight ω ( f ) = ∑ x ∈ V ( G ) | f ( x ) | among all outer-independent 2-rainbow dominating functions f on G . In this note, we obtain new results on the previous domination parameter. Some of our results are tight bounds which improve the well-known bounds β ( G ) ≤ γ r 2 o i ( G ) ≤ 2 β ( G ) , where β ( G ) denotes the vertex cover number of G . Finally, we study the outer-independent 2-rainbow domination number of the join, lexicographic, and corona product graphs. In particular, we show that, for these three product graphs, the parameter achieves equality in the lower bound of the previous inequality chain.