We consider models of integer-valued height functions on shift-invariant planar graphs whose maximum degree is three. We prove delocalisation for a large class of such models. This provides a simplified proof of the Kosterlitz-Thouless phase transition, and is, to the knowledge of the author, the first height function delocalisation proof which is not tied to a specific model. Included are: the discrete Gaussian and solid-on-solid models, as well as the uniformly random K-Lipschitz function, on the vertices of the hexagonal lattice. This talk is based on arXiv:2012.09687.