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The dominating graph of a graph G is a graph whose vertices correspond to thedominating sets of G and two vertices are adjacent whenever their correspondingdominating sets differ in exactly one vertex. Studying properties of dominatinggraph has become an increasingly interesting subject in domination theory. Onthe other hand, median graphs and partial cubes are two fundamental graphclasses in graph theory. In this paper, we make some new connections betweendomination theory and the theory of median graphs and partial cubes. As themain result, we show that the following conditions are equivalent for everygraph $G \not \simeq C_4$ with no isolated vertex, and in particular, that thesimple third condition completely characterizes first two ones in which threeconcepts of dominating graphs, median graphs and complement of minimaldominating sets get related:  - The dominating graph of G is a median graph,  - The complement of every minimal dominating set of G is a minimal dominatingset,  - Every vertex of G is either of degree 1 or adjacent to a vertex of degree1.  As another result, we prove that the dominating graph of every graph is apartial cube and also give some examples to show that not all partial cubes ormedian graphs are isomorphic to the dominating graph of a graph. Theabove-mentioned results, as another highlight of the paper, provide novelinfinite sources of examples of median graphs and partial cubes.

On dominating graph of graphs, median graphs and partial cubes, and graphs in which complement of every minimal dominating set is minimal dominating