Geodesic
Completely symmetric configurations for $\sigma $-games on grid graphs
Journal of Algebraic Combinatorics, Tome 31 (2010) no. 4, pp. 533-545.

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Summary: The paper deals with $\sigma $-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 $\geq 3$, the answer is complete for the $\sigma ^{+}$-game, and for the $\sigma ^{ - }$-game if at least one of the sizes is even. The case $\sigma ^{ - }$, dimension $\geq 3$ and all sizes odd remains open.
Keywords: keywords sigma-games, chebychev polynomials, commutative algebra 
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     author = {Florence, Mathieu and Meunier, Fr\'ed\'eric},
     title = {Completely symmetric configurations for $\sigma $-games on grid graphs},
     journal = {Journal of Algebraic Combinatorics},
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     year = {2010},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/JAC_2010__31_4_a2/}
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Florence, Mathieu; Meunier, Frédéric. Completely symmetric configurations for $\sigma $-games on grid graphs. Journal of Algebraic Combinatorics, Tome 31 (2010) no. 4, pp. 533-545. http://geodesic.mathdoc.fr/item/JAC_2010__31_4_a2/