  The dominating graph of a graph G is a graph whose vertices correspond to the
dominating sets of G and two vertices are adjacent whenever their corresponding
dominating sets differ in exactly one vertex. Studying properties of dominating
graph has become an increasingly interesting subject in domination theory. On
the other hand, median graphs and partial cubes are two fundamental graph
classes in graph theory. In this paper, we make some new connections between
domination theory and the theory of median graphs and partial cubes. As the
main result, we show that the following conditions are equivalent for every
graph $G \not \simeq C_4$ with no isolated vertex, and in particular, that the
simple third condition completely characterizes first two ones in which three
concepts of dominating graphs, median graphs and complement of minimal
dominating sets get related:
  - The dominating graph of G is a median graph,
  - The complement of every minimal dominating set of G is a minimal dominating
set,
  - Every vertex of G is either of degree 1 or adjacent to a vertex of degree
1.
  As another result, we prove that the dominating graph of every graph is a
partial cube and also give some examples to show that not all partial cubes or
median graphs are isomorphic to the dominating graph of a graph. The
above-mentioned results, as another highlight of the paper, provide novel
infinite sources of examples of median graphs and partial cubes.
