Graph states are the backbone of measurement-based continuous-variable quantum computation. However, experimental realisations of these states induce Gaussian measurement statistics for the field quadratures, which poses a barrier to obtain a genuine quantum advantage. In this letter, we propose mode-selective photon addition and subtraction as viable and experimentally feasible pathways to introduce non-Gaussian features in such continuous-variable graph states. In particular, we investigate how the non-Gaussian properties spread among the vertices of the graph, which allows us to show the degree of control that is achievable in this approach.