Efficient domination of the orientations of a graph Original Research Article

For an orientation G of a simple graph G, NG[x] denotes the vertex x together with all those vertices in G for which there are arcs directed toward x. A set S of vertices of G is an efficient dominating set (EDS) of G provided that IFld |NG[x]∩ S| = 1 for every x in G. An efficiency of G is an ordered pair (G, S), where S is an EDS of the orientation G of G. The number of distinct efficiencies of G is denoted is denoted by η(G). We give a formula for η(G) which allows us to calculate it for complete graphs, complete bipartite graphs, cycles, and paths. We find the minimum and maximum value of η(G) among all graphs with a fixed number of edges. We also find the minimum and maximum value of η(G), as well as the external 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.