The Tower of Hanoi with Forbidden Moves

We consider a variant of the classical three-peg Tower of Hanoi problem, where limitations on the possible moves among the pegs are imposed. Each variant corresponds to a di-graph whose vertices are the pegs, and an edge from one vertex to another designates the ability of moving a disk from the first peg to the other, provided that the rules concerning the disk sizes are obeyed. There are five non-isomorphic graphs on three vertices, which are strongly connected - a sufficient condition for the existence of a solution to the problem. We provide optimal algorithms for the problem for all these graphs, and find the number of moves each requires.