Diameters of Permutation Groups on Graphs and Linear Time Feasibility Test of Pebble Motion Problems

Let $G$ be an $n$-vertex connected, undirected, simple graph. The vertices of $G$ are populated with $n$ uniquely labeled pebbles, one on each vertex. Allowing pebbles on cycles of $G$ to rotate (synchronous rotations along multiple disjoint cycles are permitted), the resulting pebble permutations form a group $\G$ uniquely determined by $G$. Let the diameter of $\G$ (denoted $diam(\G)$) represent the length of the longest product of generators (cyclic pebble rotations) required to reach an element of $\G$, we show that $diam(\G) = O(n^2)$. Extending the formulation to allow $p \le n$ pebbles on an $n$-vertex graph, we obtain a variation of the (classic) pebble motion problem (first fully described in Kornhauser, Miller, and Spirakis, 1984) that also allows rotations of pebbles along a fully occupied cycle. For our formulation as well as the (classic) pebble motion problem, given any start and goal pebble configurations, we provide a linear time algorithm that decides whether the goal configuration is reachable from the start configuration. This gives a positive answer to an open problem raised by (Auletta et al., 1999)