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We study the scaling behavior of the size of minimum dominating set (MDS) inscale-free networks, with respect to network size $N$ and power-law exponent$\gamma$, while keeping the average degree fixed. We study ensembles generatedby three different network construction methods, and we use a greedy algorithmto 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 ofthe network classes display a transition at $\gamma \approx 1.9$, with linearscaling above, and vanishingly weak dependence below, but in the third networkclass we find linear scaling irrespective of $\gamma$. We find that the partialMDS, which dominates a given $z<1$ fraction of nodes, displays essentially thesame scaling behavior as the MDS.

Minimum Dominating Sets in Scale-Free Network Ensembles