Efficient twin domination in generalized De Bruijn digraphs

A twin dominating set in a digraph D=(V,A)D=(V,A) is a set S⊆VS⊆V such that, for every w∈V∖Sw∈V∖S, there exist arcs (u,w),(w,v)∈A(u,w),(w,v)∈A with u,v∈Su,v∈S. A twin dominating set SS is efficient if there is no arc in the subdigraph induced by SS and, for every w∈V∖Sw∈V∖S, there exist exactly one vertex u∈Su∈S and exactly one vertex v∈Sv∈S such that arcs (u,w),(w,v)∈A(u,w),(w,v)∈A. This paper resolves a conjecture regarding an efficient twin domination set in generalized De Bruijn digraphs. The conjecture says that the generalized De Bruijn digraphs GB(n,d)GB(n,d) with nn a multiple of d+1d+1 have an efficient twin dominating set if and only if dd is even and relatively prime to nn. This paper affirms this conjecture.