To clarify, I'm speaking of homeomorphisms in the graph theoretic context, obtained by subdivisions of the arcs of a directed graph. A subdivision of an arc $(x,z)$ in a directed graph is obtained by removing $(x,z)$ from the arc-set and adding both $(x,y)$ and $(y,z)$. This is the definition in [BJG2009; page 10]; intuitively, we add a vertex $y$ different from all existing vertices in the "middle" of our arrow, and then give it an arrow-"head" that is pointed in the same direction as the other "head":

                 [![enter image description here][1]][1]

Now, while studying the automorphism groups of digraphs. I.e., if $G=(V,E)$ is our digraph,  

> $\text{Aut}(G)=\{\sigma\in \text{Sym}(V):\ \forall x,y\in V \qquad (x,y)\in E\iff (\sigma(x),\sigma(y))\in E\ \}$.  

I've noticed a number of relationships between homeomorphisms and automorphisms. For example, if $G$ is a *finite*<sup>(*)</sup> directed acyclic graph, then all orbits of the action of automorphism group on the vertex-set are anti-chains of the poset $(V,E^{+})$, where ${}^+$ denotes transitive closure. Also, if we call a vertex $v\in V$ "thin" iff  $\text{deg}^{+}(v)=\text{deg}^{-}(v)=1$ while also referring to any path $P\subseteq G$ as a "thin path" iff every vertex of $P$ is a thin vertex in $G$, then every automorphism of $G$ permutes the maximal thin paths of like length in $G$. I.e., for each $\ell\in\omega$, there is a permutation action of $\mathrm{Aut}(G)$ on the set $\mathsf{Thin}_\ell(G)$ of all thin directed length-$\ell$-paths in $G$.

Now if we subdivide a single arrow in each of the maximal thin paths of some fixed length in $G$ it seems the order of the newly formed automorphism group is the same order as $\text{Aut}(G)$, in fact I'm pretty sure not only is this true but so are a number of other possible identities similar to this, for example here are two other propositions I'm confident to state (though I haven't formally proved them):

 1. Any *finite*<sup>(**)</sup> digraph is homeomorphic to another digraph which has a trivial automorphism group.
 2. Two directed acyclic graphs are isomorphic iff their barycentric     subdivisions are isomorphic. Or more generally iff $\forall n\in \mathbb{N}$ their $n^{\text{th}}$ barycentric subdivisions are isomorphic.

  [1]: https://i.sstatic.net/KxWLK.png

With that in mind, are there other propositions of this kind that relate the automorphism groups of homeomorphic digraphs with one another, or equivalently, relate their isomorphism classes? 

If anyone could point me to some writings that make such connections, that would be helpful as well.

${}$________________________


<sup>(*)</sup>The following statement is obviously false for infinite digraphs.

<sup>(**)</sup>Without assuming finiteness, this is false: the two-way infinite directed path (i.e., the digraph $(\mathbb{Z},<)$) has the property that *any* subdivision still has the same, and hence nontrivial, automorphism group.

**[BJG2009]** Jørgen Bang-Jensen, Gregory Gutin, *Digraphs: Theory, Algorithms and Applications*, Springer Monographs in Mathematics, Second Edition, 2009, ISBN-13: 978-0857290410