On the {2}-domination number of rooted product graphs

Abstract

Let G be a nontrivial graph with vertex set V (G). A function f : V (G) → {0, 1, 2} is called a {2}-dominating function on G if ∑ _{u ∈NG} [v] f(u) ≥ 2 for every v ∈ V (G), where N_G [v] represents the closed neighborhood of vertex v ∈ V (G). The {2}-domination number of G is the minimum weight ω(f) = ∑ _{u ∈V (G)} f(u) among all {2}-dominating functions f on G. In this paper, we obtain a closed formula for the {2}-domination number of rooted product graphs. In particular, we show that in this product graph, this parameter can attain only six possible values, which depend on some domination parameters of the graphs involved in the product. We also characterize the rooted product graphs that satisfy each of these six expressions.