Independent perfect domination sets in Cayley graphs (Q2746203)

Let \(G\) be a graph with vertex set \(V(G)\). A domination set \(S\) is called perfect if every vertex of \(V(G)-S\) is adjacent to exactly one vertex of \(S\). In the first part of the paper a necessary and sufficient condition for the existence of a decomposition of \(V(G)\) into independent perfect domination sets is given. Conditions for the existence of independent domination sets in Cayley graphs are presented in the second part. As an application of the first two parts an characterization of hypercubes with perfect independent domination set is given. The existence of an independent perfect domination set in \(Q_{n }\) is equivalent to the existence of a regular covering of the complete graph \(K_{n+1}\) by the hypercube \(Q_{n}\).