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A configuration of the lit-only $\sigma$-game on a finite graph $\Gamma$ isan assignment of one of two states, on or off, to all vertices of $\Gamma.$Given a configuration, a move of the lit-only $\sigma$-game on $\Gamma$ allowsthe player to choose an on vertex $s$ of $\Gamma$ and change the states of allneighbors of $s.$ Given any integer $k$, we say that $\Gamma$ 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 $\Gamma$ is a tree with a perfectmatching. We show that $\Gamma$ is 1-lit and any tree obtained from $\Gamma$ byadding a new vertex on an edge of $\Gamma$ is 2-lit.

Two-lit trees for lit-only sigma-game