Minimum light number of lit-only σ-game on a tree

Let T be a tree with ℓ leaves. Each vertex of T is assigned a state either lit or off. An assignment of states to all the vertices of T is called a configuration. The lit-only σ-game allows the player to pick a lit vertex and change the states of all its neighbours. We prove that for any initial configuration one can make a sequence of allowable moves to arrive at a configuration in which the number of lit vertices is no greater than . We also give examples to show that the bound cannot be relaxed to .