Domination Graphs of Tournaments and Digraphs

The domination graph of a digraph has the same vertices as the digraph with an edge between two vertices if every other vertex loses to at least one of the two. Previously, the authors showed that the domination graph of a tournament is either an odd cycle with or without isolated and/or pendant vertices, or a forest of caterpillars. They also showed that any graph consisting of an odd cycle with or without isolated and/or pendant vertices is the domination graph of some tournament. This paper extends these results to oriented graphs. We also show that any caterpillar is the domination graph of some digraph, but a path on four or more vertices is not the domination graph of any tournament. Other results relate the domination graph of a tournament to its positive subtournament deened by Fisher and Ryan, and the possible and average number of edges in the domination graph of a tournament. Let an n-digraph be an oriented graph on n vertices. Vertex x beats vertex y if there is an arc from x to y. For a vertex x, let O(x) (the out-neighbors or outset of x) denote the set of vertices which x beats, and let I (x) (the in-neighbors or inset of x) denote the set of vertices which beat x. Let d + (x) = jO(x)j and d ? (x) = jI(x)j denote the out-degree and in-degree of x. An n-tournament is an oriented complete graph. See Moon 7] and Reid and Beineke 8] for more about tournaments. Two vertices x and y dominate an n-digraph D if x and y beat all other vertices i.e., fx; yg O(x) O(y) = V (D) where V (D) is the set of vertices of D. Two such vertices are called a dominant pair. The domination graph of D, denoted dom(D), is a graph on the vertices of D with edges between vertices which are dominant pairs. Domination graphs were introduced in connection with competition graphs of tournaments. The domination graph of a tournament T is the complement of the competition graph of the reversal of T 1]. See Lundgren 5] for more about competition graphs and Lundgren, et al. 6] for more on the competition graphs of tournaments.