  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.
