Let $X$ be the dyadic rationals in the half-open unit interval.
The graph $G$ over $X$ having the vertices $(x,x+2^{\nu_2(x)-1})$ and $(x,x-2^{\nu_2(x)-1})$ is connected. It's just the infinite binary rooted tree and the proof it's connected is that it maps each number to its square in the Prufer 2-group, which is torsion.
Suppose we didn't know a priori that $G$ is connected, what exchanges of pairs of elements would need to be defined in the spirit of this question in order to define a subgroup of the permutation group, which acts transitively on $X$ if and only if $G$ is connected?
The motivation is for me to understand better, by virtue of an example I understand, how the transitivity property is being used in that question.
Is it as simple as exchanging the two ends of each edge?