We geometrically describe the relation induced on a set of graphs by isomorphism of their associated graph C*-algebras as the smallest equivalence relation generated by five types of moves. The graphs studied have finitely many vertices and finitely or countably infinitely many edges, corresponding to unital and separable C*-algebras.

The first and second named authors were supported by the Danish National Research Foundation through the Centre for Symmetry and Deformation (DNRF92). The second named author was further supported by the DFF-Research Project 2 "Automorphisms and Invariants of Operator Algebras", no. 7014-00145B. He also acknowledges support from the Rise network H2020-MSCA-RISE-2015-691246-QUANTUM DYNAMICS as indeed significant work was done on the paper while he was seconded by the network. The third named author was supported by a Simons Foundation Collaboration Grant, # 567380.

During the initial phases of this work the first named author was a postdoctoral fellow at the Mittag-Leffler Institute. All authors thank the Institute and its staff for the excellent working conditions provided.