  The $\gamma$-graph of a graph $G$ is the graph whose vertices are labelled by
the minimum dominating sets of $G$, in which two vertices are adjacent when
their corresponding minimum dominating sets (each of size $\gamma(G)$)
intersect in a set of size $\gamma(G)-1$. We extend the notion of a
$\gamma$-graph from distance-1-domination to distance-$d$-domination, and ask
which graphs $H$ occur as $\gamma$-graphs for a given value of~$d \ge 1$. We
show that, for all $d$, the answer depends only on whether the vertices of $H$
admit a labelling consistent with the adjacency condition for a conventional
$\gamma$-graph. This result relies on an explicit construction for a graph
having an arbitrary prescribed set of minimum distance-$d$-dominating sets. We
then completely determine the graphs that admit such a labelling among the
wheel graphs, the fan graphs, and the graphs on at most six vertices. We
connect the question of whether a graph admits such a labelling with previous
work on induced subgraphs of Johnson graphs.
