  We study the problem of planning paths for $p$ distinguishable pebbles
(robots) residing on the vertices of an $n$-vertex connected graph with $p \le
n$. A pebble may move from a vertex to an adjacent one in a time step provided
that it does not collide with other pebbles. When $p = n$, the only collision
free moves are synchronous rotations of pebbles on disjoint cycles of the
graph. We show that the feasibility of such problems is intrinsically
determined by the diameter of a (unique) permutation group induced by the
underlying graph. Roughly speaking, the diameter of a group $\mathbf G$ is the
minimum length of the generator product required to reach an arbitrary element
of $\mathbf G$ from the identity element. Through bounding the diameter of this
associated permutation group, which assumes a maximum value of $O(n^2)$, we
establish a linear time algorithm for deciding the feasibility of such problems
and an $O(n^3)$ algorithm for planning complete paths.
