Dominance in a Cayley digraph and in its reverse

Let D be a digraph. Its reverse digraph, D−1, is obtained by reversing all arcs of D. We show that the domination numbers of D and D−1 can be different if D is a Cayley digraph. The smallest groups admitting Cayley digraphs with this property are the alternating group A4 and the dihedral group D6, both on 12 elements. Then, for each n ≥ 6 we find a Cayley digraph D on the dihedral group Dn such that the domination numbers of D and D−1 are different, though D has an efficient dominating set. Analogous results are also obtained for the total domination number.