  We introduce a technique called graph fission which takes in a graph which
potentially contains only one observation per node (whose distribution lies in
a known class) and produces two (or more) independent graphs with the same
node/edge set in a way that splits the original graph's information amongst
them in any desired proportion. Our proposal builds on data fission/thinning, a
method that uses external randomization to create independent copies of an
unstructured dataset. We extend this idea to the graph setting where there may
be latent structure between observations. We demonstrate the utility of this
framework via two applications: inference after structural trend estimation on
graphs and a model selection procedure we term "graph cross-validation".
