On Alon's problem concerning the difference between connected domination number and domination number

For a connected graph $G$, let $γ(G)$ and $γ_c(G)$ denote its domination number and connected domination number, respectively. Let $M(n,k)$ be the maximum of $γ_c(G)-γ(G)$ over all connected $n$-vertex graphs of minimum degree at least $k$. Alon proved that \[ 2\left\lfloor\frac{n}{k+1}\right\rfloor