Trivalent graph isomorphism problem

Given data
We are given two graphs $$X_1 = (V_1, E_1)$$ and $$X_2 = (V_2,E_2)$$, with the promise that both of them are trivalent (viz the degree of every $$v \in V_i$$ is bounded by 3).

Goal
We are required to determine whether $$X_1$$ and $$X_2$$ are isomorphic as graphs.

Variant for which we solve the problem
The variant of the trivalent graph isomorphism problem that we solve here is the following: Given a trivalent graph $$X$$, and an edge $$e$$ of $$X$$, find a small generating set for $$Aut_e(X)$$, viz the graph automorphisms of $$X$$ which fix the edge $$e$$.

Here's how the trivalent graph isomorphism problem reduces to the edge-fixing trivalent graph automorphism problem. Take the two graphs $$X_1$$ and $$X_2$$. Fix a $$v \in V(X_1)$$. For each $$w \in V(X_2)$$, consider the graph