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.

图G的\（\ gamma \） -图是其顶点由G的最小支配集标记的图，其中两个顶点在其对应的最小支配集（每个大小为\（\ gamma（G ）\））相交于一组大小\（\ gamma（G）-1 \）中。我们将\（\ gamma \）-图的概念从距离1-统治到距离d-统治，并问给定值 \（d，哪些图H作为\（\ gamma \）-图出现。\ ge 1 \）。我们表明，对于所有d，答案仅取决于H的顶点是否接受与常规\（\ gamma \）图的邻接条件一致的标签。该结果依赖于具有任意规定的最小距离d控制集的图形的显式构造。然后，我们在轮图，扇形图和最多六个顶点的图之间完全确定允许这种标记的图。我们将图是否允许这样的标记与之前关于Johnson图的诱导子图的工作联系起来。