Generating the symmetric group by three prefix reversals

The cubic pancake graphs are Cayley graphs over the symmetric group Symₙ generated by three prefix reversals. There is the following open problem: characterize all the sets of three prefix reversals that generate Symₙ. As the largest prefix reversal of length n is always included in a triple, we give a complete solution of the problem when any of the two smallest or the two largest lengths but n are included in a triple of prefix reversals. Moreover, some conditions implying a triple of prefix reversals does not generate Symₙ are considered. Computational results on the diameter and the girth of some cubic pancake graphs are presented, and conjectures for future research are formulated. 26 pages, 4 tables, 3 figures, 27 references