Finding a set which dominates the Minimum Dominating Set
The algorithm previously proposed in the post (before the current edit), does not work.

Here are the problem and algorithm proposed in that version of the post:

input: graph $G=(V,E)$ and the minimum size $k$ of any dominating set in $G$

output: a set of $k$ nodes that dominates a dominating set of size $k$


  algorithm$(G,k)$:
  
  At the start, all nodes are unmarked.
  
  1) Pick an unmarked node $v$ as a center and mark everything within 2
  hops of $v$.  [This step was edited and differs in the current post.]
  
  2) Repeat step 1 until all nodes are marked. (Note, this will never
  place more than $k$ centers)
  
  3) If we have picked less than $k$ centers, keep placing centers on
  undominated nodes until we have placed $k$ centers.





  Lemma. The algorithm above doesn't work.


Proof. Consider the counter-example $G=(V,E)$ where 


$V=\{a, b, c, d, e, f\}$
$E=\{(a, b), (a, c), (a, d), (b, e), (c, f), (d, f)\}.$




The (unique) minimum dominating set is $\{b, f\}$, with size $k=2$.

The algorithm can place a center on $v=a$ in Step 1 (marking all nodes), then place a center on $e$ in Step 3.  This returns a set $\{a, e\}$ of two centers, neither of which dominates $f$ (which is in the only dominating set of size $2$). $~~~\Box$