Generating the symmetric group by three prefix reversals

The cubic pancake graphs are Cayley graphs over the symmetric group ${\mathrm{S}\mathrm{y}\mathrm{m}}_n$ generated by three prefix reversals. There is the following open problem: characterize all the sets of three prefix reversals that generate ${\mathrm{S}\mathrm{y}\mathrm{m}}_n$. We present a partial answer to this problem, in particular, we characterize all generating sets of three elements that contain at least one of the prefix reversals $r_2,r_3,r_{n-2}$, and $r_{n-1}$. We also give some computational results relating to the diameter and the girth of some cubic pancake graphs.