I'm trying to prove the following. Let G be a strongly connected subgraph of a de Bruijn graph on the whole k-ary alphabet (there is a directed path from every vertex to every other). Let X be a set of cycles in G with the following closure property: if the sequence (vertex) v is in X, then X also contains either all the edges departing from v, or it contains the inverse of v and all edges departing from it.

Show that X must be equal to G.