By a result of Landau (1953), we know that every tournament has a king (a vertex from which every vertex is reachable by a path of length at most 2). Let T be a tournament such that δ - (T) ≥ 1, that...

By a result of Landau (1953), we know that every tournament has a
king (a vertex from which every vertex is reachable by a path of length at most 2). Let T be a tournament such that δ^-(T) ≥ 1, that is, d^-(v) ≥ 1 for all v ∈ V (T).

1. Show that if x is a king in T, then T has another king in N^-(x).

2. Using the answer to the previous question, prove that T has at least 3 kings.

3. For each n ≥ 3, give a construction of a tournament T' with n vertices such that δ^-(T') ≥ 1 and T' has exactly 3 kings.