On graphs with a unique minimum hull set

We show that for every integer k ≥ 2 and every k graphs G1, G2, . . . , Gk, there exists a hull graph with k hull vertices v1, v2, . . . , vk such that link L(vi) = Gi for 1 ≤ i ≤ k. Moreover, every pair a, b of integers with 2 ≤ a ≤ b is realizable as the hull number and geodetic number (or upper geodetic number) of a hull graph. We also show that every pair a, b of integers with a ≥ 2 and b ≥ 0 is realizable as the hull number and forcing geodetic number of a hull graph.