Consider asymmetric unlabelled digraphs $G$ (asymmetric means |Aut($G$)| = 1).

Trivially, (i) each node $v$ can be uniquely labelled (with the pointed graph $G_v$, i.e. $G$ with distinguished node $v$), and (ii) whether there's an edge between $v$ and $w$ can be read off their labels $G_v$ and $G_w$.

Things get interesting when in the case of an (countably) infinite graph $G$ a finite induced subgraph of $G_v$ for all $v$ does suffice for (i) and (ii). Let's call such a graph and its nodes *finitely discriminable*.

There are two obvious examples of infinite finitely discriminable graphs:

the natural number graph with edges from $n$ to $n+1$ in which each node $v$ can be labelled by the induced subgraph consisting of $v$ and its predecessors

the graph of hereditarily finite sets with edges from $x$ to $y$ iff $x \in y$ in which each node $v$ can be labelled by its transitive closure graph TC({v})

What these two graphs do share are the Mostowski collapse conditions, i.e. their relation is set-like, well-founded and extensional.

Questions:

Are the Mostowski conditions necessary and/or sufficient for an infinite graph to be finitely discriminable?

If they are not necessary: what is an example of an infinite finitely discriminable graph for which one of them does not hold? Is the rest of them necessary, then?

Especially: Are there undirected (i.e. not well-founded) finitely discriminable graphs?

If the Mostowski conditions are not sufficient: can they be augmented to become sufficient, or is there another completely different set of sufficient conditions?