Completely symmetric configurations for σ-games on grid graphs
The paper deals with σ-games on grid graphs (in dimension 2 and more) and conditions under which any completely symmetric configuration of lit vertices can be reached – in particular the completely lit configuration – when starting with the all-unlit configuration. The answer is complete in dimension 2. In dimension ≥3, the answer is complete for the σ +-game, and for the σ −-game if at least one of the sizes is even. The case σ −, dimension ≥3 and all sizes odd remains open.