  In this paper, we give a series of couterexamples to negate a conjecture and
hence answer an open question on the $k$-power domination of regular graphs
(see [P. Dorbec et al., SIAM J. Discrete Math., 27 (2013), pp. 1559-1574]).
Furthermore, we focus on the study of $k$-power domination of claw-free graphs.
We show that for $l\in\{2,3\}$ and $k\ge l$, the $k$-power domination number of
a connected claw-free $(k+l+1)$-regular graph on $n$ vertices is at most
$\frac{n}{k+l+2}$, and this bound is tight.
