2 00 9 Difference between minimum light numbers of sigma - game and lit - only sigma - game ∗

A configuration of a graph is an assignment of one of two states, on or off, to each vertex of it. A regular move at a vertex changes the states of the neighbors of that vertex. A valid move is a regular move at an on vertex. The following result is proved in this note: given any starting configuration x of a tree, if there is a sequence of regular moves which brings x to another configuration in which there are l on vertices then there must exist a sequence of valid moves which takes x to a configuration with at most l+ 2 on vertices. We provide example to show that the upper bound l+2 is sharp. Some relevant results and conjectures are also reported.