Graph wavelet transforms allow for the effective representation of signals that are defined over irregular domains. The transform coefficients should be sparse, and encode salient features of a signal. In many situations, these salient features appear as discontinuities in the signal, e.g. physical edges in natural images. The transforms facilitate the development of various graph signal processing tasks, e.g. feature extraction or compression. The graph wavelets proposed in the literature are linear transforms, and employ linear filters that consist essentially of mean operations. We propose the construction of nonlinear graph wavelets that are based on the nonlinear median operator. The construction process converts existing linear transforms that are based on the critically sampled two-channel graph filter bank. There are two parts to the conversion process. The first part derives a lifting structure similar to that found in classical wavelet transforms. Given a pair of polynomial filter functions representing the filter bank, a polyphase factorization technique, that is based on the Euclidean algorithm for polynomials, will be developed. The resulting lifting structure enables a more modular design of graph wavelet transforms, but is still linear. The second part of the process, termed medianfication, converts certain linear operators, that are associated with the graph shift, into corresponding median operators. The resulting nonlinear transform shares many desirable properties of the corresponding linear transform. The nonlinear transform has the perfect reconstruction property. Though strictly nonlinear, the transform has similar spectral characteristics to the linear counterpart. It is able to separate low-pass features, which are smooth approximation of a signal, from high-pass features, which are rapid variations in a signal. A graph transform has the edge-aware property if filtering across signal discontinuities is avoided, and this leads to better perfor...