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