The number of dominating $k$-sets of paths, cycles and wheels

We give a shorter proof of the recurrence relation for the domination polynomial $\gamma (P_n,t)$ and for the number ${\gamma }_k(P_n)$ of dominating $k$-sets of the path with $n$ vertices. For every positive integers $n$ and $k,$ numbers ${\gamma }_k(P_n)$ are determined solving a problem posed by S. Alikhani in CID 2015. Moreover, the numbers of dominating $k$-sets ${\gamma }_k(C_n)$ of cycles and ${\gamma }_k(W_n)$ of wheels with $n$ vertices are computed.