Generalized switch-setting problems

Switch-setting games like Lights Out are typically modelled as a graph, where the vertices represent switches and lamps, and the edges capture the switching rules. We generalize the concepts used for a mathematical description of Lights Out and its relatives. Our approach uses bipartite graphs and allows for the analysis of a broader class of switch-setting games, which is demonstrated at a new variant of the Lights Out puzzle. Our method exhibits full duality between switches and lamps, and we get rid of some insufficiencies inherent in the modelling with non-bipartite graphs. We present a detailed analysis of the new Lights Out variant, formulate solvability conditions, give graphically aesthetic interpretations and discuss aspects on minimum solutions. In our study of parity domination in bipartite graphs we incorporate methods from linear algebra and linear programming. We point out the close relations between graph theoretic terms and the language of algebra over Z2Z2.