Which graphs occur as $γ$-graphs?

The $γ$-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 $γ(G)$) intersect in a set of size $γ(G)-1$. We extend the notion of a $γ$-graph from distance-1-domination to distance-$d$-domination, and ask which graphs $H$ occur as $γ$-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 $γ$-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.

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