Game theory on graphs is used in AI for adversarial search in game-playing agents (chess, Go) and in multi-agent systems where agents compete or cooperate.
Minimax, alpha-beta pruning, and Grundy numbers are tools in AI game solvers. Modern AI uses neural networks to approximate minimax values, but the underlying theory comes from graph game analysis. If you understand backward induction on graphs, you understand the foundation behind all adversarial AI.