  In this paper, we first show that the power domination number of a connected
$4$-regular claw-free graph on $n$ vertices is at most $\frac{n+1}{5}$, and the
bound is sharp. The statement partly disprove the conjecture presented by
Dorbec et al. in SIAM J. Discrete Math., 27:1559-1574, 2013. Then we present a
dynamic programming style linear-time algorithm for weighted power domination
problem in trees.
