The {2}-domination in rooted product graphs

A \(\{2\}\)-dominating function (\(\{2 \}\)DF) on a graph \(G=(V(G),E(G))\) is a function \(f : V(G) → \{0,1,2 \}\) such that \(f(N[v]) ≥ 2\) for every \(v ∈ V(G)\), where \(N[v]\) is the closed neighourhood of \(v\). The \(\{2\}\)-domination number of \(G\) is the minimum weight \(ω(f) = ∑v ∈ V(G) f(v)\) among all \(\{2 \}\)-dominating functions on \(G\). In this article, we prove that if \(G\) and \(H\) are graphs with no isolated vertex, then for any vertex \(v ∈ V(H)\) there are six closed formulas for the \(\{2\}\)-domination number of the rooted product graph \(G ∘_v H\). We also characterize the graph \(G\) and \(H\) that satisfy each of these formulas.