Graphs with large double domination numbers

In a graph G, a vertex dominates itself and its neighbors. A subset S ⊆ V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number ${\displaystyle {\gamma }_{\times 2}\left(G\right)}$. If G ≠ C₅ is a connected graph of order n with minimum degree at least 2, then we show that ${\displaystyle {\gamma }_{\times 2}\left(G\right)\le 3\frac{n}{4}}$ and we characterize those graphs achieving equality.