On the outer-independent {2}-domination in rooted product graphs

Abstract

Let G be a graph with vertex set V ( G ). A function f : V ( G ) → {0, 1, 2} is called an outer-independent {2}-dominating function on G if ∑ _{u ∈ N [ v ]} f ( u ) ≥ 2 for every vertex v ∈ V ( G ) and { v ∈ V ( G ) : f ( v ) = 0} is an independent set of G . The minimum weight ω ( f ) = ∑ _{v ∈ V ( G )} f ( v ) among all outer-independent {2}-dominating functions on G is the outer-independent {2}-domination number of G . In this article, we investigate this parameter for rooted product graphs. In particular, we obtain a closed formula consisting of four distinct expressions and characterize the graphs for which each expression applies. As a consequence, we also obtain the corresponding closed formula for corona product graphs.