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The Domination-Compliance Graph of a Tournament

Vertices x and y are a dominant pair in a tournament if for all vertices z not equal to x,y, either x beats z or y beats z. Vertices x and y are a compliant pair in a tournament if for all vertices z not equal to x,y, either z beats x or z beats y. Let DC(T) be the graph on the same vertex set as T with edges between pairs of vertices that are either a dominant pair or a compliant pair in T. We show that the maximum possible number of edges in DC(T) is 2(n-1) and this bound is sharp. In addition we obtain results about the structure of DC(T) such as forbidden subgraphs and the clique number. Since DC(T) is the complement of the Competition/Resource graph of a tournament, complementary results are obtained for this graph.