Efficient domination of the orientations of a graph

For an orientation G of a simple graph G, -Ni7 [x] denotes the vertex x together with all those vertices in ~ for which there are arcs directed toward x. A set S of vertices of ~ is an efficient dominating set (EDS) of G provided that [IqS[x]nS [ = 1 for every x in G. An efficiency of G is an ordered pair (~, S), where S is an EDS of the orientation ~ of G. The number of distinct efficiencies of G is denoted by t/(G). We give a formula for t/(G) which allows us to calculate it for complete graphs, complete bipartite graphs, cycles, and paths. We find the minimum and maximum value of t/(G) among all graphs with a fixed number of edges. We also find the minimum and maximum value of t/(G), as well as the extremal graphs, among all graphs with a fixed number of vertices. Finally, we show that the probability a random oriented graph has an EDS is exponentially small when such graph is chosen according to a uniform distribution.