The four-valued Belnap–Dunn logic is a well-established non-classical logic which provides one way of formally reasoning with inconsistent and incomplete information. The basic idea is simple and natural: allow for the possibility that a formula may be both true and false (if we have inconsistent information) or neither true nor false (if we have incomplete information). Apart from its three-valued cousins, however, virtually nothing is known about the extensions of this logic. Our aim in this talk will therefore be to investigate the rich landscape of logics lying between the Belnap–Dunn logic and classical logic. It turns out that from the semantic point of view, these logics have a graph-theoretic interpretation, whereas from the proof-theoretical point of view, they are obtained from classical logic by dropping or relaxing the cut and identity rules. In particular, we may use these non-classical logics to prove cut and identity elimination for classical logic.