Two-lit trees for lit-only sigma-game

A configuration of the lit-only $σ$-game on a finite graph $Γ$ is an assignment of one of two states, on or off, to all vertices of $Γ.$ Given a configuration, a move of the lit-only $σ$-game on $Γ$ allows the player to choose an on vertex $s$ of $Γ$ and change the states of all neighbors of $s.$ Given any integer $k$, we say that $Γ$ is $k$-lit if, for any configuration, the number of on vertices can be reduced to at most $k$ by a finite sequence of moves. Assume that $Γ$ is a tree with a perfect matching. We show that $Γ$ is 1-lit and any tree obtained from $Γ$ by adding a new vertex on an edge of $Γ$ is 2-lit.

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