On double domination in graphs

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 ${\displaystyle {\gamma }_{\times 2}\left(G\right)}$. A function f(p) is defined, and it is shown that ${\displaystyle {\gamma }_{\times 2}\left(G\right)=minf\left(p\right)}$, where the minimum is taken over the n-dimensional cube ${\displaystyle Cⁿ=\{p=(p₁,...,pₙ)\mid p_i\in IR,0\le p_i\le 1,i=1,...,n\}}$. Using this result, it is then shown that if G has order n with minimum degree δ and average degree d, then ${\displaystyle {\gamma }_{\times 2}\left(G\right)\le \left(\frac{\ln (1+d)+\ln \delta +1}{\delta }\right)n}$.