This paper revisits the problem of consistency maintenance when insertions or deletions are performed on a valid database containing marked nulls. This problem comes back to light in real-world Linked Data or RDF databases when blank nodes are associated to null values. This paper proposes solutions for the main problems one has to face when dealing with updates and constraints, namely update determinism, minimal change and leanness of an RDF graph instance. The update semantics is formally introduced and the notion of core is used to ensure a database as small as possible (\ie the RDF graph leanness). Our algorithms allow the use of constraints such as tuple generating dependencies (tgd), offering a way for solving many practical problems.