We prove neighbourhood canonicity and strong completeness for the logics EK and ECK, obtained by adding axiom (K), resp. adding both (K) and (C), to the minimal modal logic E. In contrast to an earlier proof in , ours is constructive. More precisely, we construct minimal characteristic models for both logics and do not rely on compactness of first order logic. The proof involves a specific circumscription technique and quite some set-theoretic maneuvers to establish that the models satisfy the appropriate frame conditions. After giving both proofs, we briefly spell out how they generalize to four stronger logics and to the extensions of the resulting six logics with a global modality.