I have recently been interested in the following purely graph-theoretic notion that weakens the assumption of transitivity in a similar way to how quasi-crystals have "(possibly) aperiodic long-range order" instead of exact periodicity. It is related to the notion of a minimal shift in topological dynamics (but is defined without reference to a group action) and I think can reasonably be thought of as a topological counterpart of the measure-theoretic notion of a unimodular random rooted graph. I'm interested whether this notion has ever been studied before, and if so under what name?
Here is the definition: Let $\mathbb{G}$ be the space of isomorphism classes of connected, locally finite rooted graphs (i.e., with a distinguished root vertex). The space $\mathbb{G}$ has a natural topology, sometimes called the local topology or Benjamini-Schramm topology, where each rooted graph $(G,o)$ has a basis of open neighbourhoods $N_r(G,o):=\{(H,\rho) :$ the ball of radius $r$ in $(H,\rho)$ is isomorphic to the ball of radius $r$ in $(G,o)\}$.
For each (isomorphism class associated to a) rooted graph $(G,o)$, let $[(G,o)]$ be the set of (isomorphism classes of) rooted graphs obtained by changing the root vertex of $(G,o)$, and let $\overline{[(G,o)]}$ be the closure of this set in $\mathbb{G}$. We say that a set $A \subseteq \mathbb{G}$ is re-rooting invariant if $[(G,o)] \subseteq A$ for every $(G,o)\in A$.
The notion I am interested in is minimal closed re-rooting invariant subsets of $\mathbb{G}$, that is, closed, re-rooting invariant subsets of $\mathbb{G}$ that do not admit any strict non-trivial subsets that are also closed and re-rooting invariant.
If $(G,o)$ is a transitive graph then $\{(G,o)\}$ is a minimal closed re-rooting subset of $\mathbb{G}$. Similarly, if $(G,o)$ is quasi-transitive (meaning that the action of its automorphism group on its vertex set has finitely many orbits) then we can obtain an associated minimal closed re-rooting invariant subset of $\mathbb{G}$ with finitely many elements. More interesting examples arise from aperiodic tile sets (like the Penrose tiling), which naturally lead to uncountable minimal closed re-rooting invariant subsets of $\mathbb{G}$.
In the absence of a better name, one could define a rooted graph $(G,o)$ to be pseudotransitive if $\overline{[(G,o)]}$ is a minimal closed, re-rooting invariant subset of $\mathbb{G}$. Thus, transitive and quasi-transitive graphs are pseudotransitive, but so are certain graphs associated to the Penrose tiling that are not quasi-transitive.
Edit: For a simple example of a graph that is not pseudotransitive, simply add a pendant edge to the origin of a transitive graph. The closure $\overline{[(G,o)]}$ contains the original transitive graph, and the singleton set contains this graph is a smaller closed re-rooting invariant set.
Has anyone studied this notion before?
