  We study the scaling behavior of the size of minimum dominating set (MDS) in
scale-free networks, with respect to network size $N$ and power-law exponent
$\gamma$, while keeping the average degree fixed. We study ensembles generated
by three different network construction methods, and we use a greedy algorithm
to approximate the MDS. With a structural cutoff imposed on the maximal degree
($k_{\max}=\sqrt{N}$) we find linear scaling of the MDS size with respect to
$N$ in all three network classes. Without any cutoff ($k_{\max}=N-1$) two of
the network classes display a transition at $\gamma \approx 1.9$, with linear
scaling above, and vanishingly weak dependence below, but in the third network
class we find linear scaling irrespective of $\gamma$. We find that the partial
MDS, which dominates a given $z<1$ fraction of nodes, displays essentially the
same scaling behavior as the MDS.
