Two-lit trees for lit-only sigma-game

TL;DR

This paper investigates the lit-only sigma-game on finite trees with perfect matchings, proving such trees are 1-lit and that adding a vertex on an edge increases the property to 2-lit.

Contribution

It establishes that trees with perfect matchings are 1-lit and that adding a vertex on an edge makes the tree 2-lit, advancing understanding of the game's configurations.

Findings

Trees with perfect matchings are 1-lit.

Adding a vertex on an edge makes the tree 2-lit.

The results classify trees based on their lit-only sigma-game properties.

Abstract

A configuration of the lit-only $\sigma$-game on a finite graph $\Gamma$ is an 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 $\Gamma$ allows the player to choose an on vertex $s$ of $\Gamma$ and change the states of all neighbors 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 perfect matching. We show that $\Gamma$ is 1-lit and any tree obtained from $\Gamma$ by adding a new vertex on an edge of $\Gamma$ is 2-lit.