The flipping puzzle on a graph

Let S be a connected graph which contains an induced path of n-1 vertices, where n is the order of S. We consider a puzzle on S. A configuration of the puzzle is simply an n-dimensional column vector over {0,1} with coordinates of the vector indexed by the vertex set S. For each configuration u with a coordinate us=1, there exists a move that sends u to the new configuration which flips the entries of the coordinates adjacent to s in u. We completely determine if one configuration can move to another in a sequence of finite steps.