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 u s = 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.