Efficient Open Domination

A set S of vertices of a graph G is called an efficient open domination set for G if the set of neighborhoods {N(v)⫫ v ϵ S} forms a partition of V(G). A graph is an efficient open domination graph if it contains an efficient open domination set. Several properties of efficient open domination sets and efficient open domination graphs are determined. All efficient open domination trees are described. The efficient open domatic number of a graph is defined and studied.