  We formalize eight different notions of isomorphism among (unital) graph
C*-algebras, and initiate the study of which of these notions may be described
geometrically as generated by moves. We propose a list of seven types of moves
that we conjecture has the property that the collection of moves respecting one
of six notions of isomorphism indeed generate that notion, in the sense that
two graphs are equivalent in that sense if and only if one may transform one
into another using only these kinds of moves.
  We carefully establish invariance properties of each move on our list, and
prove a collection of generation results supporting our conjecture with an
emphasis on the gauge simple case. In two of the six cases, we may prove the
conjecture in full generality, and in two we can show it for all graphs
defining gauge simple C*-algebras. In the two remaining cases we can show the
conjecture for all graphs defining gauge simple C*-algebras provided that they
are either finite or have at most one vertex allowing a path back to itself.
