Natural hazards such as floods, heatwaves and windstorms can cause havoc for the people affected and typically result in huge financial losses. Drug-induced liver injury is a major public health and industrial issue and is typically triggered by a combination of extremes of laboratory variables. There is a clear need to have good methods to estimate the likelihoods of extreme events to help in mitigating the risk. Motivated by the structure observed in limit laws of conditional extremes of stationary Markov processes, we develop graphical modelling methodology that allows dimension reduction and efficient high-dimensional inference for multivariate extremes. A statistical procedure is described and efficiency gains are illustrated via simulation.