Completely symmetric configurations for σ-games on grid graphs

Abstract

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. © Springer Science+Business Media, LLC 2009.