Some general aspects of the framing number of a digraph

A digraph D is homogeneously embedded in a digraph H if for each vertex x of D and each vertex y of H, there exists an embedding of D in H as an induced subdigraph with x at y. A digraph F of minimum order in which D can be homogeneously embedded is called a frame of D and the order of F is called the framing number of D. Several general results involving frames and framing numbers of digraphs are established. The framing number is determined for a number of classes of digraphs, including a class of digraphs whose underlying graph is a complete bipartite graph, a class of digraphs whose underlying graph is C_n + K₁, and the lexicographic product of a transitive tournament and a vertex transitive digraph. A relationship between the diameters of the underlying graphs of a digraph and its frame is determined. We show that every tournament has a frame which is also a tournament.