Lights Out on graphs

We model the Lights Out game on general simple graphs in the framework of linear algebra over the field ${\mathbb{F}}_2$. Based upon a version of the Fredholm alternative, we introduce a separating invariant of the game, i.e., an initial state can be transformed into a final state if and only if the invariant of both states agrees. We also investigate certain states with particularly interesting properties. Apart from the classical version of the game, we propose several variants, in particular a version with more than only two states (light on, light off), where the analysis resides on systems of linear equations over the ring ${\mathbb{Z}}_n$. Although it is easy to find a concrete solution of the Lights Out problem, we show that it is NP-hard to find a minimal solution. We also propose electric circuit diagrams to actually realize the Lights Out game.