Luks began with a trivalent graph $X$ with $|V(X)| = n$ and we want to determine the $Aut_{e}(X)$ (automorphism with a fixed edge $e$) by a sequence of what he calls "approximations". He breaks the graph in subgraphs $X_{r}$ of $X$ where all vertices and edges of $X_{r}$ appear in some path of length of size $r$ or less through the edge $e$.
We have that $X_{1}$ is just $e$ and $X_{n-1}$ is exactly the original graph $X$. Then, he built the induced homomorphism $\pi_{r} : Aut_{e}(X_{r+1}) \rightarrow Aut_{e}(X_{r})$ and assumed that $Aut_{e}(X_{r})$ is already known. Then, the determination of $Aut_{e}(X_{r+1})$ is broken up into two problems:
- Find a set $\Re$ of generators for the kernel $K_{r}$ of $\pi_{r}$;
- Find a set $\Im$ of generators for $\pi_{r}(Aut(X_{r+1})) = Image(\pi_{r})$.
Then, he claims that we can build the generators of $Aut_{e}(X_{r+1})$ with $\Re \cup \Im'$, where $\Im'$ is the pullback of the set $\Im$ ($\pi_{r}(\Im') = \Im$) and that finding the set of generators $\Re$ is easy. I want to know why $\Re \cup \Im'$ generates $Aut_{e}(X_{r+1})$ and how can we calculate the set of generators $\Re$?
EDIT 1: Luks Algorithm Paper
Here, the kernel (generated by $\Re$) can be thought of as the "new" automorphisms, i.e. those which move vertices in $V(X_{r+1}) \setminus V(X_r)$ when $X_r$ is fixed. In other words, all the possible combinations of swaps among twins.
The image (generated by $\Im'$) can be thought of as those "old" automorphisms (i.e. the ones inherited from $X_r$) which remain compatible with the new constraints.
Mixing both corresponds to playing with both degrees of freedom simultaneously, that is, $Aut_e(X_{r+1})$.
Since the kernel corresponds to all combinations of swaps among twins, it is sufficient to consider as generators the set of individual swaps, i.e., the "transpositions in each pair of twins" (top of page 49, journal version).