  Every countable graph can be built from finite graphs by a suitable infinite
process, either adding new vertices randomly or imposing some rules on the new
edges. On the other hand, a profinite topological graph is built as the inverse
limit of finite graphs with graph epimorphisms.
  We propose to look at both constructions simultaneously. We consider
countable graphs that can be built from finite ones by using both embeddings
and projections, possibly adding a single vertex at each step. We show that the
Rado graph can be built this way, while Henson's universal triangle-free graph
cannot.
  We also study the corresponding profinite graphs. Finally, we present a
concrete model of the projectively universal profinite graph, the projective
Fraisse limit of finite graphs, showing in particular that it has a dense
subset of isolated vertices.
