The worst Hanoi graphs

The “Towers of Hanoi” is a problem that has been extensively studied and frequently generalized. We are interested in its generalization to arbitrary directed graphs and ask how many moves are required in a given graph to move n disks from the starting peg to the destination peg. Not all directed graphs allow solving this problem; we will call those graphs that do Hanoi graphs. We settle the question of what are the Hanoi graphs that require the largest number of moves.