Describe the automorphism group of the digraphs

Question

Describe the automorphism group of the digraphs

Here is what I got so far

For $D_1$, because of the direction of $vw$, we can't do anything to the graph so $Aut(D_1)= I$

For $D_2$, we can flip the graph horizontally, so we have 2 automorphism, the identity and $\alpha=(u w)(c x)$

For $D_3$ I think the only think we can do is flip the graph over the arc $uy$, so we have 2 automorphisms the identity and $\alpha=(u x)$

I wonder if I missed or misunderstand anything?

Answer 1

The two first are right, but for the third one you're missing some -- for example the automorphism that swaps $x$ and $y$ and leaves the other three nodes alone. In fact there are 6 automorphisms including the identity.

Note that a graph isomorphism doesn't need to be a geometric symmetry of the drawing of the graph -- it must just preserve the abstract which-nodes-have-edges-between-them relation.