Counting lines in semi-complete digraphs *

A digraph D = (V, A) is semi-complete if for each pair of distinct vertices x and y in V, either xy or yx belong to A. A subset ℓ of vertices is a line of D if there are two distinct vertices x and y such that for any vertex z ε V, z ε ℓ if and only if a directed shortest path exists containing x, y and z.