Say that a digraph $(V,E)$ is *reducible* if there exist $x,y\in V$ with $x\ne y$ and such that for all $z\in V$, $(x,z)\in E\leftrightarrow(y,z)\in E$ and $(z,x)\in E\leftrightarrow(z,y)\in E$. It is clear that a reducible digraph has a non-trivial automorphism (swapping $x$ and $y$ and fixing the rest), but the inverse is not necessarily true.


For some applications elsewhere, I'm interested in "irreducible" digraphs and especially "irreducible" posets. Now, (insert usual disclaimer about not being an expert), do these already have a (potentially better) name? and if yes, where have they been treated?



Digraph without "immediately isomorphic" vertices?
Just to provide a conclusion here (summing up the comments): *twin-free* seems to be the best and, dare I say, standard terminology; two points as in my question would then be called *twins*. Otherwise, *point-determining*, *R-thin*, or *mating-type* have been used to denote the same property.


