Graphical models of extremes reveal how EU emissions-allowance futures connect to other markets during unusually large price movements. The analysis covers twenty daily variables across the third and fourth phases of the EU Emissions Trading System, from 2013 to 2025, and compares extreme dependence with average dependence. The extreme-event networks are denser and organized around different central variables. Allowance futures become especially central under stress, suggesting that models based only on average relationships can miss important channels of transition risk.