The efficient k-distance domination number γke(G) is the minimum among the cardinalities of efficient k-distance dominating sets of G. Upper bound in terms of order and maximum degree of an independent set S of vertices in a graph G=(V,E) called an efficient k-distance dominating set if every vertex in V−S be distance k from exactly one vertex in S has been presented. Sufficient conditions for 3-regular Cayley graphs to have disjoint efficient k-distance dominating sets are given. Characterization of the 3-regular connected circulant graphs that admit an efficient k-distance dominating set for k=2,3 has been obtained.
