Consider the following method of creating a graph $G'(V',E')$ from a graph $G(V,E)$:
An edge of G becomes a vertex of G' Two verteces $v', w'\in V'$ are connected if the edges they come from are connected. That is, if $e_1,e_2\in E$ are such that $\exists u,v,w\in V\ :\ e_1=(u,v), e_2=(v,w)$
That way an isolated cycle stays the same kind of cycle. An isolated edge becomes an isolated vertex.
It's always possible to do this forward but its not always possible to find the "inverse" of a graph.
If we keep finding the correspondant G', G''etc it can either stabilize, disapear or diverge into an infinitely big graph.