  A (simple) hypergraph is a family H of pairwise incomparable sets of a finite
set. We say that a hypergraph H is a domination hypergraph if there is at least
a graph G such that the collection of minimal dominating sets of G is equal to
H. Given a hypergraph, we are interested in determining if it is a domination
hypergraph and, if this is not the case, we want to find domination hypergraphs
in some sense close to it, the domination completions. Here we will focus on
the family of hypergraphs containing all the subsets with the same cardinality,
the uniform hypergraphs of maximum size. Specifically, we characterize those
hypergraphs H in this family that are domination hypergraphs and, in any other
case, we prove that the hypergraph H is uniquely determined by some of its
domination completions and that H can be recovered from them by using a
suitable hypergraph operation.
