Lit-only sigma-game on nondegenerate graphs

TL;DR

This paper characterizes the behavior of the lit-only sigma-game on nondegenerate, non-line graphs, showing they are 2-lit and providing criteria for 1-lit status using linear algebra.

Contribution

It offers a complete description of the game dynamics on certain graphs and introduces a linear algebraic condition for 1-lit graphs, expanding understanding of the game.

Findings

Nondegenerate, non-line graphs are 2-lit.

Provided a linear algebraic criterion for 1-lit graphs.

Described the orbits of the game on these graphs.

Abstract

A configuration of the lit-only $\sigma$-game on a graph $\Gamma$ is an assignment of one of two states, {\it on} or {\it off}, to each vertex of $\Gamma.$ Given a configuration, a move of the lit-only $\sigma$-game on $\Gamma$ allows the player to choose an {\it on} vertex $s$ of $\Gamma$ and change the states of all neighbors of $s.$ Given an integer $k$, the underlying graph $\Gamma$ is said to be $k$-lit if for any configuration, the number of {\it on} vertices can be reduced to at most $k$ by a finite sequence of moves. We give a description of the orbits of the lit-only $\sigma$-game on nondegenerate graphs $\Gamma$ which are not line graphs. We show that these graphs $\Gamma$ are 2-lit and provide a linear algebraic criterion for $\Gamma$ to be 1-lit.