The collection of the vertex dominating sets of a graph defines a hypergraph on the set of vertices of the graph. However, there are hypergraphs H that are not the collection of the vertex dominating sets of any graph. This paper deals with the question of completing these hypergraphs H to the vertex dominating sets of some graphs G. We demonstrate that such graphs G exist and, in addition, we prove that these graphs define a poset whose minimal elements provide a decomposition of H. Moreover, we show that the hypergraph H is uniquely determined by the minimal elements of this poset. The computation of such minimal elements is also discussed in some cases.