The transitive closure of a set *X* is usually seen as a set, but it can also be seen as a graph *G(X)* with V(*G*)= TC(*X*) and (*x,y*) ∈ E(*G*) iff *x* ∈ *y*. Such a (transitive closure) graph reveals irredundantly everything that is to know about the set ("*its hidden ∈-structure*"). It is known that *G(X)* ≅ *G(Y)* iff *X* = *Y*.

*G(X)* obviously 

 1. contains exactly one vertex with no out-arrows
 2. contains no two vertices with the same parents (→ axiom of extensionality)
 3. contains no directed loops (→ axiom of foundation)

Is this enough to characterize the class of transitive closure graphs of hereditarily finite sets: 

> Is there a 1:1 correspondence between
> the (isomorphism types of) finite
> digraphs with properties (1)-(3) and
> $V_\omega$?