An iterative strategy for Lights Out on Petersen graphs

Abstract

We establish some preliminary results for Sutner’s ${\sigma }^{+}$ game, known as Lights Out, played on the generalized Petersen graph $P\left(n,k\right)$. While all regular Petersen graphs admit game configurations that are not solvable, we prove that every game on the $P\left(2n,n\right)$ graph has a unique solution. Moreover, we introduce a simple iterative strategy for finding the solution to any game on $P\left(2n,n\right)$, and generalize its application to a wider class of graphs.