This is a contradiction since D is a tournament. Now suppose that a tournament D has a unique Hamiltonian path. Let a m be the first vertex in this path that has an arc coming to it from a higher numbered vertex. Let a k be the largest numbered vertex that has an arc going to a m . So, in all cases, we obtain a contradiction.
Every acyclic digraph has a vertex with no incoming arcs.
D' is acyclic since D was. Take a path of maximal length in D'. Consider the last vertex, v, in this path. There is no arc from v to any other vertex in the path since D' is acyclic. There is no arc from v to any other vertex since the path was of maximal length.