We present characterizations of connected graphs G of order n ≥ 2 for which h + ( G ) = n . It is shown that for every two integers n and m with 1 ≤ n − 1 ≤ m ≤ ( n 2 ) , there exists a connected graph G of order n and size m such that for each integer k with 2 ≤ k ≤ n , there exists an orientation of G with hull number G .