The worst parallel Hanoi graphs

The “Towers of Hanoi” is a problem that has been extensively studied and frequently generalized. In particular, it has been generalized to be played on arbitrary directed graphs and using parallel moves of two types. We ask what is the largest number of parallel moves, in either of the two models, that is required to move n disks from the starting node to the destination node. Not all directed graphs allow solving this problem; we will call those graphs that do Hanoi graphs. In previous work, we settled the question of what are the worst sequential Hanoi graphs, that is, those graphs that require the largest number of sequential moves. We also demonstrated that the characterization of sequential Hanoi graphs carries over the parallel Hanoi graphs. Here, we determine the worst Hanoi graphs provided parallel moves are allowed. It turns out that for one of the two models of parallel moves, the worst graphs are quite different from the worst sequential graphs, while in the other model of parallelism, there is little difference with the sequential situation.