We show how any labeled convex polygon associated to a compact semi-toric system, as de fined by V (u) over tilde Ngoc, determines Karshon's labeled directed graph which classifies the underlying Hamiltonian S-1-space up to isomorphism. Then we characterize adaptable compact semi-toric systems, i.e. those whose underlying Hamiltonian S-1-action can be extended to an effective Hamiltonian T-2-action, as those which have at least one associated convex polygon which satisfies the Delzant condition.