Perfect directed codes in Cayley digraphs

A perfect directed code (or an efficient twin domination) of a digraph is a vertex subset where every other vertex in the digraph has a unique in- and a unique out-neighbor in the subset. In this paper, we show that a digraph covers a complete digraph if and only if the vertex set of this digraph can be partitioned into perfect directed codes. Equivalent conditions for subsets in Cayley digraphs to be perfect directed codes are given. Especially, equivalent conditions for normal subsets, normal subgroups, and subgroups to be perfect directed codes in Cayley digraphs are given. Moreover, we show that every subgroup of a finite group is a perfect directed code for a transversal Cayley digraph.